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Bifurcations and stability of families of diffeomorphisms

Bifurcations and stability of families of diffeomorphisms We consider one parameter families or arcs of diffeomorphisms. For families starting with Morse-Smale diffeomorphisms we characterize various types of (structural) stability at or near the first bifurcation point. We also give a complete description of the stable arcs of diffeomorphisms whose limit sets consist of finitely many orbits. Uni- versal models for the local unfoldings of the bifurcating periodic orbits (especially saddle- nodes) are established, as well as several results on the global dynamical structure of the bifurcating diffeomorphisms. Moduli of stability related to saddle-connections are introduced. CONTENTS CI~R I. -- Introduction ................................................................. 7 CHAPTER II. -- Local description of the elementary bifurcations ............................ 13 I. Introduction ......................................................................... 13 2. Center manifolds and periodic elementary bifurcations ................................... 13 3- The saddle-node ..................................................................... 15 4. The fllp ............................................................................. 2x 5. The Hopf point ...................................................................... 22 6. Quasi-transversal intersections .......................................................... ~ CHAPaXR III. -- Necessary conditions for stability of ares .................................. 26 i. The modulus condition (quasi-transversal intersection) .................................... 26 2. Necessary conditiom for mild stability and stability ...................................... 37 3. Endomorphisrm of the circle .......................................................... 43 4. The saddle-node with l-cycle in dimension 2 ........................................... 48 5. On the rigidity of the unfolding of the saddle-node ...................................... 5x 5 S. NEWHOUSF_,, J. PALIS AND F. TAKENS CHaR IV. -- Global stability ............................................................ 55 I. Introduction ......................................................................... 55 2. Local tubular families ................................................................ 55 3. Global tubular families ............................................................... 59 4. Stability ............................................................................ 64 This research was carried out mainly at the Instituto de MatemCttica Pura e Aplicada, Rio de Janeiro. The first two authors were also supported by an agreement between GNPq (Brazil) and NSF (USA). I. -- INTRODUCTION The study of the geometric structure of the orbits of dynamical systems (differential equations, flows, vector fields, or diffeomorpkisms) defined on a manifold has been considered in many works since Poincar~ and Liapunov. Two diffeomorphismsfandf' are said to have the same geometric structure if they axe topologically conjugate, i.e. if there is a homeomorphism h from the domain off to that off' such that hf =f' h. Two flows or vector fields, are called topologically equivalent if there is a homeomorpkism sending orbits of one system to orbits of the other; if, in addition, the homeomorphism preserves the flow parameter, we again say the systems are conjugate. In general terms, we aim at the classification of dynamical systems under conjugacy or topological equi- valence. Since, however, much pathological behavior can occur, we must restrict ourselves to interesting special classes of systems. We shall be concerned here with systems having only mild recurrence; in particular, we shall frequently assume that their limit sets consist of only finitely many orbits. The space of differentiable systems of class C r, r > I, has a natural topology given by uniform convergence of the first r derivatives. This is called the C' topology. Given any equivalence relation E on the set of dynamical systems, one can define systems to be E-stable if they lie in the interior of their E-equivalence classes. When topological equivalence is used for E, an E-stable system is called structurally stable (or just stable). The stable diffeomorphisms and vector fields whose limit sets have finitely many orbits coincide with the Morse-Smale ones [24]. In fact, in this case the Birkhoff center [I5] is finite because non-trivial recurrence implies uncountably many orbits in the limit set. Since one understands the structure of Morse-Smale systems pretty well, it is natulal to consider one-parameter families of systems starting at a Morse-Smale one and to attempt to describe the structure of the elements of such families. In the present work we shall define three natural equivalence relations on these families, and we shall characterize their stable families in terms of geometric properties. In particular, we will characterize the stable one parameter families of diffeomorphisms whose elements have only finitely many orbits in their limit sets. This corresponds to the characterization of the Morse- Smale diffeomorphisms as the ones which are stable and have finitely many orbits in their limit sets. The results can, of course, be translated to certain classes of vector fields (those with global cross-sections). In [26a] related results are obtained for families of gradient vector fields. We first present a preliminary description of our main results. Later, we shall give their precise statements. S. NEWHOUSE, J. PALIS AND F. TAKENS In [I9] , [20], one studied how a one-parameter family of diffeomorphisms starting at a Morse-Smale one ceases to be stable (i.e. goes through a bifurcation) when the parameter evolves. For a generic family, the description given there is complete assuming that the diffeomorphism at the frst bifurcation point has its limit set made of a finite number of orbits. It turns out that these orbits are all periodic except at most one. If the periodic orbits are all hyperbolic, then their stable and unstable manifolds meet transversally except along one orbit. In case one of the periodic orbits is not hyperbolic, this orbit must be an elementary bifurcation (a saddle-node, flip, or Hopf orbit), the other periodic orbits must be hyperbolic, and all stable and unstable manifolds meet transversally. The orbit structure of the diffeomorphism at the first bifurcation point will be basic for our results on the stability of these arcs. Throughout this paper, M denotes a compact C ~~ manifold without boundary, and Diff(M) denotes the set of C | diffeomorphisms of M. We let 9 ~ ---- 9~(M) be the space of C ~~ arcs ofdiffeomorphisms on M. That is, if I is the unit interval, then ~(M) consists ofG ~176 mappings 9 : M � I ~ M � I, such that ~(m, ~) = (q~(m), ~) where m ~ ~,(m) is a G ~~ diffeomorphism for each Ez ~ I. Elements of ~ will also be called one-parameter families or arcs of diffeomorphisms and will frequently be denoted by { %,} or q~. We give Diff(M) and ~ the usual &o topologies. Let us consider three equivalence relations on the set ~ of one-parameter families of diffeomorphisms. We say two families are topologically conjugate if, modulo an orien- tation preserving homeomorphism of the interval I, each element of the first family is topologically conjugate to an element of the second family, and the conjugacy varies continuously with the parameter. If conjugacies exist but do not necessarily vary conti- nuously, we say the families are mildly conjugate. Finally, if the elements of the families are topologically conjugate up to and including their first bifurcation points, we say they are left conjugate. The interiors of the equivalence classes of the preceding equivalence relations define, respectively, stable, mildly stable, and left stable arcs of diffeomorphisms. For arcs (~} such that the limit set of r consists of finitely many orbits for each ~z, we will give necessary and sufficient conditions for stability, and we will geometrically characterize left stability. For mild stability, our characterization is complete except for one case which we present as an interesting open question. In aU cases, we exhibit necessary and sufficient conditions in terms of the orbit structures at the bifurcation points of the arcs. For arcs beginning at Morse-Smale diffeomorphisms we will first study the different types of stability for an interval in I containing the first bifurcation point in its interior. Let us describe the results. To begin with, if, at the first bifurcation point, some stable and unstable manifolds meet non-transversally, then the arc is not even left stable. In fact, in this case we have at least a one-parameter family of different equivalence classes of arcs near the initial one. This corresponds to the existence of the modulus of stability as discussed in [27], [28]. In the other possible cases, when there is a saddle- node, a flip or a Hopf periodic orbit, the arc is left stable. When the arc goes through BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS a Hopf orbit it is never mildly stable; this is due to the appearance of invariant circles with irrational rotations. On the other hand, we always get stability for arcs going through flip orbits. The case of a saddle-node orbit deserves a special discussion. First of all, it is much harder than in the flip case to prove the existence up to conjugacy of a universal model for its local unfolding. This is done in Chapter II. On the other hand, the restriction of such a conjugacy to the center manifold is surprisingly rigid. We discuss several applications of this fact in section 5 of Chapter III. Moreover, the strong stable and strong unstable foliations of the stable and unstable manifolds must be preserved by a conjugacy between two arcs going through saddle-nodes. Thus, criticaUity or tangency of the invariant (stable, unstable) manifolds of other periodic orbits with respect to one of these foliations implies that the arc is not stable. CriticaUity with respect to both foliations impfies that the arc is not even mildly stable. Another crucial factor is the existence of cycles for the periodic orbits. When the arc goes through a saddle-node which is critical but not bicritical and has no cycles, then this arc is mildly stable but not stable. If there is a cycle, the arc is not stable and, we believe, it is not even mildly stable. We are able to prove this last statement for cycles of length bigger than one and for one- cycles when the saddle-node is normally attracting or repelling. This follows from the appearance of a non-transversal homoclmic orbit, which implies a non zero modulus of stability. To prove the existence of such a homoclinic orbit, we reduce the question to one-parameter families of endomorphisms of the circle and introduce a generalized notion of rotation number. Necessary conditions for the types of stability mentioned above are also established in Chapter III. The woof that, under these conditions, the arcs are stable, mildly or left stable is performed in the last chapter. There we use a suitable version of tubular families or foliations, some of them with singularities. Our constructions also provide a more elegant woof of the stability of Morse-Smale diffeo- morphisms originally established in [24], [25]. Stability for arcs containing saddle-nodes was generalized by Robinson [31 ] to certain families starting at Axiom A diffeomorphisms. Many of these results were announced in [2I]; however, we mistakenly claimed to have characterized mild stability. As we mentioned above, it remains to prove that certain /-cycle cases are not mildly stable. Let us now review some definitions and be more precise. For g e Diff(M), the orbits(x), of a point x ~ M, is defined as ,(x) = {g"(x) [ n ~ Z}. A point y E M is called a limit point of g if for some sequence n~ ~ Z with [ nil -~ 0% limg'~(x) =y. We denote by L(g) the closure ofthe set of these limit points. A point x ~ M is a periodic point ofg with period k if gk(x) = x and gt(x) 4= x for all o < t < k; x is hyperbolic if dgk(x) has no eigenvalues on the unit circle. The stable, unstable sets or manifolds W'(x, g), W~(x, g), of a periodic point x are defined as {y ~ M [ p(g"(y), g"(x)) -+ o for n ~ + o0} and {y c M l o(g"(Y), g"(x)) -+ o for n ~ -- oo}, 2 IO s. NEWHOUSE, J. PALIS AND F. TAKENS respectively, where p is a metric on M. Ifx is a hyperbolic periodic point of g, W'(x, g) and W"(x, g) are injectively immersed sub-manifolds of M. We say that a diffeomorphism g 9 Diff(M) with finitely many periodic orbits has a k-cycle if there is a sequence of periodic orbits ~(P0), ..., r(Pk) with ~(P0) = ~(Pk) and r(Pi+t) Cclosure (WU(~(pi)) for o < i < k, and r(p~) 4= r(pj) for o < i < j < k, and if this sequence is maximal. A diffeomorphism g is Morse,Smale if L(g) is finite and hyperbolic and if all the intersections of stable and unstable manifolds are transverse. We denote the set of Morse-Smale diffeomorphisms by MS. This set is open [24] and each g e MS is stable in the sense that any g' which is C t near g is conjugate to g [24], [25], i.e. there is a homeo- morphism h : M ~ M so that g'h = hg. For an arc {q~,} ~ with q~0 eNIS, let b = b(~) = inf{~t eI [ q~, CMS}. We always assume that b(~) < I. If {%,},{9'~,} 9 thenwesay that (h,{H~}) is aconjugacy if h : [o, I] ---> [o, I] is a homeomorphism with h(b(~)) =b(~?'), H~,:M~M is a conjugacy between q~, and 9~/~,)for all ~t in some neighborhood of [o, b(~)], and H,, depends continuously on ~t. If H~, does not necessarily depend continuously on ~t, we say that (h, {Hr,}) is a mildconjugacy, and ifH~, is only a conjugacy for ~z < b(q~), not necessarily continuous in V, then (h, {H~,}) is called a left eonjugaey. Clonjugacy, mild conjugacy and left conjugacy define equivalence relations in the set of those arcs in ~ which start in MS. An arc (~,} e ~ is called stable, mildly stable or left stable if is an interior point of its corresponding equivalence class. Now we come to the description of the class of arcs to which our results apply. Definition.- z~r is the subset of those {q~.} e~ such that I. q~o e MS; 2. b = b(q~) = inf{~t ~[o, t] I ~ r I; 3. the limit set of ~ has finitely many orbits. For the arcs of diffeomorphisms in ~ it is often useful to impose certain generic conditions on the first bifurcation. In order to describe them we need some more conditions. Let x be a fixed point ofa diffeomorphism g e Diff(M). We call x quasi-hyperbolic if one of the following three conditions holds: -- (dg)x has one eigenvalue one, the other eigenvaiues have norm different from I and there is a g-invariant curve = through x such that g J - has first but not second order contact with the identity at x; in this case, x is a saddle-node; -- (dg)x has one eigenvalue -- x, the other elgenvalues have norm different from I and there is a g-invariant curve ~ through x such that g2 [ = has second but not third order contact with the identity at x; in this case, x is a flip. -- (dg), has a pair X 4= X of eigenvalues on the unit circle, the other eigenvalues have norm different from i and there is a g-invariant surface ~ through x, tangent to the 10 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS x z generalized eigenspace of the pair 7,, ). at x and such that the 3-jet ofg at x makes g [ = an attractor or a repeller; in this case, x is a Hopf point. For periodic orbits there is an analogous definition of quasi-hyperbolicity. When {%} ~ .~g, we say that (%} is elementary at its first bifurcation value b (or % is elementary) if it fulfills one of the two following conditions: (I) all periodic points of % are hyperbolic, there is one orbit of non-transversal intersections of a stable and an unstable manifold, and all other intersections of stable and unstable manifolds are transversal; or (2) there is one quasi-hyperbolic orbit, the other periodic orbits are hyper- bolic and all intersections of stable and unstable manifolds are transverse. In case x is a flip or saddle-node of ~b we also require stable and unstable manifolds to be transversal to the strong stable and unstable manifold of x; the strong stable (resp. unstable) manifold consists of the points y such that the distance from x to q~(y) (resp. ~-~(y)) goes exponentially to zero (see also Chapter IV). If {%} ~ ~, ~b is elementary, and % has a quasi-hyperbolic periodic orbit, then there are generic conditions one may impose on the dependence upon tz at these quasi- hyperbolic periodic points. Such conditions are described in Chapter II, w 3 (for the saddle-node), w 4 (for the flip) and w 5 (for the Hopf orbit). If these conditions are satisfied we say that the quasi-hyperbolic orbit unfolds generically. Definition. -- ,~C ~t' is defined to be the set of those arcs {%} in a~ for which % is elementary and for which the quasi-hyperbolic periodic point of %, if there is any, unfolds generically. It can be shown [4], [x9], [2o], [3 6] that there is a residual subset M' in ~ such that int(~) = d ng'. We want to give necessary and sufficient conditions for arcs in g to be stable, mildly stable or left stable. For this we need to describe the notion of criticallity. Let g e Diff(M) and let x be a saddle-node ofg. Then there is a unique foliation o~ TM of W'(x, g) with smooth leaves such that the boundary of W~ g) is a leaf and such that g maps leaves to leaves; see [I 2]. ~"' is called the strong stable foliation. A corresponding foliation of WU(x, g), the strong unstable foliation, is denoted by .~-". We call x s-critical if there is some hyperbolic periodic point y of g such that W"(y, g) intersects some leaf of ~"~ non-transversally; u-critical is defined similarly. Now, x is called semi-critical if it is either s- or u-critical, x is called bi-critical if it is both s- and u-critical, and x is called non-critical if it is not semi-critical. Theorem. -- Let { ~} be an arc in ~. I. { ~} is left stable if and only if all stable and unstable manifolds of ~ intersect transversally; 2. if{%) is left stable, if the quasi-hyperbolic orbit is not a bi-critical saddle-node or a Hopf orbit and if ~b has no cycles, then { %} is mildly stable; 11 I2 S. NEWHOUSE, J. PALIS AND F. TAKENS 2'. /f{%} is mildly stable, then {%} is left stable, the quasi-hyperbolic orbit is not a bi-critical saddle-node or a Hopf orbit, % has no cycles of length greater than i, and ~b has no non-critical 1 -cycles; 3. { ~} is stable if and only if{ %,} is left stable, the quasi-hyperbolic orbit of ~b is not a semi-critical saddle-node or a Hopf orbit and ?b has no cycles. This theorem, together with the remark at the end of section 2, Chapter III, and Theorem (4.4) in Chapter IV implies the following characterization of stable arcs of diffeomorphisms with limit sets consisting of finitely many orbits. Theorem. -- Let { q~ }, Vt ~ [o, I ] be an arc of diffeomorphisms such that the limit set of each ~?~ consists of finitely many orbits, ~ ~ [o, i]. Then { ~} is stable if and only if there are onlyfinitely many bifurcation values, say bx, ..., b,, in (o, I) and for each i < i < s, ~bl has the following properties: -- all stable, strong stable, unstable, and strong unstable manifolds intersect transversally; -- ~bi has no cycles and has exactly one non-hyperbolic periodic orbit which is either a flip or a non-critical saddle-node; this non-hyperbolic periodic orbit unfolds generically. We acknowledge useful comments by several colleagues, especially W. de Melo, F. Przytycki, and C. Pugh. 12 II. -- LOCAL DESCRIPTION OF THE ELEMENTARY BIFURCATIONS I. Introduction Let M be a smooth manifold. We shall consider smooth arcs of diffeomorphisms ?~:M-+M, ~t~R, i.e. such that the corresponding map ~:M�215 defined by ~(x, Ez) = (~(x), ~) is C ~. For such an arc, there usually are points (x, ~t) E M � R where % does not satisfy the Kupka-Smale conditions [i3] , [33] along the orbit of ~ through x. In this chapter we shall analyze the behavior of 9 near those points. Such a point is called an elementary bifurcation (point) of 9 (or ~). There are two types of elementary bifurcation points, namely those (x, ~t) for which x is a non-hyperbolic periodic point of %, and those (x, Ez) where x is a non-transversal point of intersection of stable and unstable manifolds of ~ (see [36]). Before we go into details we describe some facts concerning center manifolds in relation to the periodic elementary bifurcation points. Then a description of the types of periodic elementary bifurcations occurring in generic arcs is given. In subsequent sections all these types are analyzed. The saddle-node elementary bifurcation shows some exceptional and unexpected topological properties. In the final section, non-transversal intersections of stable and unstable manifolds are treated. 2. Center m~s,;folds and periodic elementary bifurcations Let {% : M --~ M} be a smooth arc of diffeomorphisms having a periodic bifur- cation at (~, ~). We assume ~to be a fixed point of r if not we replace ~ by 9~ where k is the period of ~-. Let c be the number of eigenvalues of (dq~); of norm I ; since (s ~) is a bifurcation, c ~ i. From the theory ofinvariant manifolds [I2], we conclude the existence ofa "center manifold depending on ~ ", namely a differentiable submanifold W e of M � R such that: -- zw'; O(W ~) n W ~ is open in W ~ (and contains (~, ~)); -- the dimension of W # is c + i and at each point (x, ~) e W ~, W ~ is transversal with respect to M � {~t}; d(~ I W ~ r3 (M X {~}))i has only eigenvalues of norm I. For any k ~ oo we may assume that W ~ is 0 ~ (but as k gets bigger, it may be necessary to take W ~ smaller). However if in some neighborhood of ~ in W~ ----- W ~ n (M � {~)), 13 x4 S. NEWHOUSE, J. PALIS AND F. TAKENS every orbit opt(x) tends to Efor i -+ 4- 0% then W e can be chosen so that W~ is C| even in this last case, W' can not be made (:1% We may consider r I We as a (local) arc of diffeomorphisms; the parameter being the restriction of ~ to W e. Also from the invariant manifold theory [26], [3 o] it follows that r near (E, ~), up to a " local conjugacy ", is completely determined by r I We and the normal data. The normal data consist of the numbers of eigenvalues of (dq~r with norm > i, resp. < i, and of the signs of the determinants of (d~)~, restricted to the maximal invariant expanding, resp. contracting subspaces of T~(M). We say that an arc r is at (~,~) locally conjugate to the arc r215 at (x, ~) if there is a homeomorphism (the local conjugacy) h from a neighborhood of (~, ~) to a neighborhood of (x, ~) such that -- h o 9 ---- ~ o h wherever defined; -- there is a local homeomorphism h a : R ~ R, defined in a neighborhood of ~ such that the ~-coordinate of h(x, ~) equals ha(~) wherever h(x, ~) is defined. So in order to analyze 9 up to local conjugacy at (~, ~) it suffices to analyze 9 [ W e up to local conjugacy; this will be done in the following sections. It should be pointed out that for 9 as above, one can choose invariant manifolds W e, and W ~', the center-stable and the center-unstable manifolds, for ~, containing W e such that the tangent space at (~, ~) is the direct sum of the tangent space of W e at (~, ~) and the maximal invariant subspace of T~(M) on which (dq~)~ has only eigenvalues with norm smaller, resp. bigger, than i. These invariant manifolds are in general not unique. In W e~ and W ~ one can choose invariant continuous foliations o~, and ~-", the strong stable and the strong unstable foliations, such that the leaves are C t, the tangent planes of the leaves depend continuously on the base-point in W% resp. W ~', and such that each leafintersects W e transversally (in W ~', resp. W ~) in one point. Also these foliations are not unique. For details on invariant manifolds and foliations see [i2] and also Chapter IV of this paper. If the leaves of o ~e', or ~-~', have co-dimension one in W~' = W e' r3{tt}, or W~" = W ~ n{~}, then, by [II], the foliation ~"', resp. o~ -"~', is C t. Let 9 and ~ be two arcs of diffeomorphisms with center manifolds W e and W e, let k : W ~ -+ W e be a local conjugacy between @ l We and 9 [ W', and let the normal data of 9 and ~ be equal. Choose invariant manifolds and foliations W e', W '~, o~ TM and o~-~, for ~, and Wee, W ~', ~" and o&" for ~. From [26] it follows that there is an extension H of h to a local conjugacy between 9 and ~ so that H(W e') -~ W e', H(W e~) = W% H(o~") = o~" and H(#-") = ~". In the case of generic arcs r the only periodic bifurcations are those of the following three types (recall we assume ~ fixed): I. c .= i, (dg~)i has art eigenvalue t and the 2-jet of r [ W~ at ~ is different from the 2-jet of the identity; in this case (E, ~.) is called a saddle-node of ~; 14 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS ]5 2. c ---= x, (d~)i has an eigenvalue --x and the 3-jet of q~[ W~ at x is different from the 3-jet of the identity; in this case we call (~, ~) a flip of ~; 3. c = 2, (dg~)i has a pair of non-real complex conjugate eigenvalues on the unit circle and the 3-jet of q~ [ W~ makes ~- an attractor or a repeller, we call these points Hopf points. A periodic bifurcation will be called elementary if it is of type i, 2, or 3 above and it unfolds generically with ~t. This last condition will be explained in sections 3, 4, and 5. Before we consider these cases in more detail, we make some more general remarks: -- in all these cases W~ is C ~~ (because near ~-, each orbit q~(x) tends to 5c as i -+ + oo or as i --* -- oo), but generically it is not possible to choose W c to be C ~~ (we shall not use or prove this fact but see [37]). This results described in this section, except those concerning saddle-nodes are all more or less well-known. Apart form the references in the various sections, one may consult [I], [4], [I9], [2o], [35], [36] 9 3. The saddle-node First we consider arcs of diffeomorphisms { %,} of R (with coordinate x) such that d d ~ d %(0) =o, aTx%(~ = I, ~%(x) 4= o and ~%(o) 4= o (to simplify notation we took ~ = o and ~-= o). Note that if ~ is a generic arc of diffeomorphisms, then ~, restricted to a center manifold of a saddle-node, has the above form. Without loss of d a d generality, we shall assume that ~-~ %(x) > o and ~ %(0) > o. We shall also assume that %(x), as a function ofx and ~, is C 6 and that %(x) is a C ~~ function ofx. Arcs of diffeomorphisms {%} satisfying the above conditions will be called (in this section) saddle-node arcs. We shall prove the following results. Theorem (3. x). -- Any two saddle-node arcs are locally conjugate near (o, o). Moreover the conjugacy can be chosen to be continuously differentiable off the fixed point set. We recall that if{%} is a saddle-node arc, there is a unique C | vector field X, defined on a neighborhood of o in R, such that the time i map X x of X equals % [39]. Theorem (3.2). -- Let (%} and {~,} be two saddle.node arcs with corresponding vector fields X and X, i.e. such that X t = % and X x = ~o. Let h be a local conjugacy between { %} and {~,}. Then h_=hl{x<o,~t=o} and h+=h[{x>o,~t=o} are G ~ and (L).x = x I{ x > o}. Remark (3.3). -- The above theorem implies that the choice of the conjugacy along {~ = o} is extremely restricted. Instead of the usual freedom to fix the conjugacy 15 16 S. NEWHOUSE, J. PALIS AND F. TAKENS arbitrarily on a fundamental domain, we are here only free to fix h in two points: one in {x<o) and one in {x>o}. Let { ~,} be an arc of diffeomorphisms of an n-dimensional manifold M having a saddle-node fixed point at (s ~). Let r t~) = (q~,(x), t~). We say that (Y, ~) unfolds generically if for some (or any) center manifold W ~ at (~-, ~), 9 [ W ~ is a saddle-node arc. Similarly, one can define generic unfolding of periodic saddle-nodes. Theorem (3.4). -- Let ~ be a smooth arc of diffeomorphisms of M, and let (~, ~) be a saddle-node of 9 which unfolds generically. Let W ~ = { (x, -~) [ O~(x, "~) ~ (~, ~) as i -~ oo } (note that W~CW ~, that dim(W') ----dim(W ~) -- I and that W * has a boundary containing (~q ~)). Then the strong stable foliation o~ TM, restricted to W ~, is unique and is preserved under any conjugacy. The proofs of these theorems occupy the rest of this section. As part of the proof of Theorem (3. x), we need to prove the corresponding theorem for vector fields. Consider vector fields X =- X(x, ~.) 0-x on It2 with X(o, o) = o, X(o, o) = o, 0 ~ 0 Ox ~ X(o, o)> o, ~ X(o, o)> o and which are at least C ~. These vector fields are called saddle-node fields. A saddle-node field can of course be considered as a one- parameter family of vector fields on R; its time one map is a saddle-node arc (except for the differentiability); see the beginning of this section. Two saddle-node fields X and X are called locally conjugate if there is a homeomorphism h (compare section 2) from a neighborhood of (o, o) in R ~ to another such neighborhood such that -- h o X t ---- X~o h, whenever defined (X t stands for the time t map of the vector field X) ; -- there is a local homeomorphism h R : It --~ It such that the ~x-coordinate of h(x, ~) equals hs(~. ) whenever defined. First we shall prove: Theorem (3-5). --Any two saddle-node fields are locally conjugate by a conjugacy which is continuously differentiable with respect to x, in the complement of the singular set. Proof of Theorem (3.5). -- Let X and X be two saddle-node fields. For X=X(x,~) 0S we choose a box V ={i x,~) I lxl<a,o<~<_~} such 0 X(x, ~) are positive on U\(o, o). Define f: (o, r ~It+ by that X(x, ~) and ~ XI(n) (- a, 8) = (-k- a, ~), or, equivalently, f(~) ----- f+a (X(x, ~))-Xdx. From this last formula it is clear that lirn0f(8 ) = -4- oo and that the derivativef'(~) is less than zero for 8 ~ (o, r Let U, f, if, ~ be defined analogously for X. Pick o< ~< r so that 16 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS i7 f(0:) 3- max(f(~),f(~)) and take o< ~< ~ so that f(~) =f(~). Then there is a homeomorphism ~q : [o, 0~] -+ [o, ~] such that f(~) =f(~(8)) for o < ~ <_ ~. A conju- gacy h from X to X, restricted to {o < ~ < ~} is now determined by putting h(- = (- a, h(a, = (a, h(o, o) = (o, o); and requiring that hX t = Xth. The continuity of h is automatic except at (o, o). We now prove continuity at (o, o). For any sequence (x~, ~) in U, there are sequences s~, t such that X~/(a, ~q) = xl and Xtl (- a, t~/) ---- xi. (x~,&) ~(o,o) if and only if Ix~--*o, s~--*---o% and t~->+oo. Hence h maps sequences converging to (o, o) to sequences converging to (o, o) which proves continuity. The construction of h [{~t < o} is easy. For example, take h(-- a, tz) = (-- if, ~), h(o, tz) = (o, ~) and h(a, ~) = (if, ix) and extend with hX~ = X~h. On the comple- Oh ment of the set of singularities of X, Ox clearly exists and equals X(h(x, Ix)). (X(x, Ix))-x which is continuous. This completes the proof of the theorem. Let (~} be a saddle-node arc, and let X be the C ~~ vector field defined near o in R such that X 1 ---- ~o. It would be useful if there were a (2 ~ field X extending X to a neighborhood of(o, o) in R' of the form X(x, ~t) = 2K(x, ix) Ox such that X(., Ix)t = %,(.). Although we cannot find such art X, lemma (3.6) provides us with a suitable substitute. We say that a (3 ~ saddle-node field X is adapted to a saddle-node arc {q~a} if the function g(x, ix), defined by (%,(x) + g(x, ~t), IX) = Xx(x , iz), vanishes along the x-axis and has at (o, o) its 4-jet equal to zero. Lemma (3.6). -- For each saddle-node arc {%,} there is an adapted saddle-node field X. Proof. -- q% is C ~~ and has, at x = o, only a finite order of contact with the identity, so by [39], ~0 embeds in a unique C| field ~ = X(x) oX" In general [I4] , [4o] if ~F : (R", o) -~ (R", o) is a C ~~ diffeomorphism with all eigenvalues of (anF)0 equal to i, there is a unique k-jet of a vector field [Y]k, k > i, such that any representative Y ~ [Y]~ satisfies -- the k-jets of Yt and q-' agree at o; -- the eigenvalues of dY at o are all equal to zero. If we apply this to (I) : (x, ~) ~ (%,(x), Ix) at (o, o) we see that the corresponding 4-jet [X] at (o, o) is uniquely determined and has a representative of the form X(x, IX) Ox" 3 I8 S. NEWHOUSE, J, PALIS AND F. TAKENS The restriction of this 4-jet to the x-axis equals the 4-jet of the previously constructed because of uniqueness of that 4-jet (which follows if we apply the above general statement 0 The to q~0 at o). X is now obtained by taking X = (X(x, bt) + X(x) -- X(x, o)) 0x" proof of (3.6) is complete. The next lemma compares high iterates of a saddle-node arc with high iterates of the time one map of an adapted saddle-node field. Lemma (3.7). -- Let{%,} be a saddle-node arc and X = X(x, ~) 0x an adapted saddle- node field which is at least G 5. Let U = { (x, ~t) I o < V. <_ -~, -- a < x < a} be so that for (x, ~) ~ U\(o, o), X(x, ~) > o, ~(x) > x, (~(-- a), ~) ~ U and ]g(x, ~) ] < ~(x) -- x. (Here g(x, Vt) is the function defined just before Lemma (3.6). The fact that the last condition is satisfied for a and Iz small enough follows from [g(x, ~)[ = o(] (x, ~t)[~) and [ 9~(x) -- x) [ > k(~ + x') on { ~t > o} for some constant k.) Then there are constants C1, ~ such that for any (x, ~t) e U, i ~ N, with (~(x), ~t) E U, (i) X~(x, V0 = (9~(x), ~t) for some o~ ~ R with I i - ~ [< ~. C1; (ii) I log((d(9~)x)X(x, ~t)) -- log(X(~(x), ~t))[ < ~.C~. Proof. -- The above two estimates on (high) iterates of ~ are implied by the following two: there are constants K1, K2 such that for any (x, ~t) e U with (~,(x), ~t) ~ U (i)' X~(x, ~t) = (?~(x), V~) for some ~ with I i -- o~ I < p.. (~(x) -- x).Kx; (ii)' [ log((d~)=.X(x, ~)) -- log(X(%,(x), ~t)) I ~ < B. (%,(x) - x).K2. Of course in (ii)' we have to assume that (x, ~) 4: o, but since for ~t = o the whole lemma is trivial we shall assume, in what follows, that always ~t > o. To show that (ii)' really implies (ii) it suffices to observe that d(r (x(x, ~)) = 'rI' d(~)~x)(X(~(x), ~)) x(r ~) ~=0 x(~,~+'(~), ~) " In the following calculation, A will indicate that the formula in which it occurs is valid ifA is replaced by some positive constant, i.e. independent of (x, ~) ~{ (x, tx) [ (x, ~t) ~ U, t~ > o, (q~(x), t~) ~ U}. From the various definitions we obtain IX(x, ~)1 _> a. (x' + ~,) and ]g(x, ~)1 < A.~.(x~ + ~')~ (here one uses that X is G~). Define h(x, ~t) by Xx+~(~)(x, ~t) = (%(x), ~). Since h(x, ix) = f~ IX(y, ~z)]-~dy, where ~ = Xx(x , ~) and [~ = @~,(x), the above inequa- lities imply I h(x, ~)1 < A. ~. (x' + ~'). 18 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS Now (i)' follows from the obvious inequality I ~(x) - x l _> A. (x~ + ~). To prove (ii)', we observe that from previous definitions it follows that (a) the 4-jet of (r -- X at (o, o) is zero (since the 4-jet of X t and r agree at (o, o) ; see Lemma (3.6)) (b) r -- X is zero along {~ = o}. Hence and I xcv~(~), ~)1 ~ ~. ((v.(~))' + ~'), I (a~L. X(x, ~) - x(~Ax), ~)l <_ A. ~. ((~A~)), + ~). so l x(~(~), ~)1 Since (x, tz) ~ (opt(x), ~) is a diffeomorphism, ((?~(x)) ~ + ~) < A(x ~ + B"). From these inequalities it easily follows that I log((do~)~. X(x, ~)) -- log(X(o~(x), ~)) [ <_ A. ~. (x s + ~) <__ A. ~. ($~(x) -- x); this proves (ii)'. Proof of Theorem (3. I). -- The proof of Theorem (3-I) will be obtained by showing that if{%,} is a saddle-node arc and X an adapted vector field (of class C 5) then there is a local conjugacy H(x, tz) = (h~,(x), ~) from {%,} to the time one map X 1 of X. We OH shall also construct H so that Ox exists and is continuous on the complement of the fixed- point set of ~. Indeed, by Theorem (3-5) and Lemma (3.6), this implies (3.1). We take h 0 = H[{~ = o} to be the identity. Let u ={(x,~)lo<_ ~_<~, --a<x<a} be as in Lemma (3.7). Take: I. h~,(--a) =--a for o<__~<~ and hence 2. (h~,(?~,(-- a)), tz) = Xn(-- a, tz) whenever (q~,(-- a), t~) e U; 3" extend the definition of h~(x) to { (x, ~) I o _< ~ _< ~, - ~ < x < ~(- a) } in such a way that it is C x and such that the extension, defined by h~(~(x), ~) = Xdh~(x), ~) is also differentiable in a neighborhood of { (0,,(-- a), t~) [ o <_ rL < g}. Now H [ U is uniquely determined; the continuity of H along { (x, ~) [ x > o, ~ = o } follows from OH Lemma (3-7); also the fact that -- is continuous on U~(o, o) follows from that same lemma. 0x 19 2O S. NEWHOUSE, J. PALIS AND F. TAKENS The extension of H to (tz < o} is simple: see the proof of Theorem (3.5). Proof of Theorem (3.2). -- Let { %,} be a saddle-node arc and X an adapted saddle- node field, of class C 5, defined in a box U as in Lemma (3.7). We take -- a < x < o and o < ~<7 a and show how {%,) gives rise to a canonical homeomorphism T~ from a neighborhood ofx (in R) to a neighborhood of~. These maps T~, are called translations. Let {xi}i~ ! be any sequence converging to x; choose a sequence {t~}~! such that ~(xi) converges to ~- with ~ -* o as i --> oo. To get such a sequence, choose ~t~ -+ o as i -+ oo such that (X~.) (x o ~q) -+ (~-, o) and apply Lemma (3.7). Let x' be a point close to x; without loss of generality we may put (x', o) -- (X~) (x, o) for some ~ e R. Then for any sequence x~ -+x', lim i ' 9 %i(x~) (the same ~q's as above) exists and equals |-+ a0 (X~)(~-, o). This can be seen as foUows: for some ~, ~, ~ we have (~(,,), ~,) = x~,(,,,, ~,); i *' t (~(~,), v,) = x~(x,, ~); x~,(x,, ~,) = (x~, ~). By Lemma (3.7) I i -- ~,1 < ~,.Cx and Ji -- ~il < re.C1 for some constant C1; because x i ~x, x; ---)-x' and X~(x, o) = (x', o), 0t i-+a. From this we conclude that if ~ (~/(xi) ~ ' , then = + ~ -- ~ so lim ~i= a. We define x~,(~,(x~), w) = v~) ~', :~, i--r oo T;.(x') lim i ' = ~(x~). It is clear that the definition of the local homeomorphism T~ is independent of X: X was only used to show that lim ~ ' i-.~o ~/(xl) exists. Observe that (Tz.).(X) = X. Now we extend the notion of translation. If x, ~- are on the same side of o and x' on the other side, we may define T;~ = T~=, o T~,=; this definition is independent (at least near x) of the choice of x'. Hence we see that the translations form the pseudo-group of all local diffeomorphisrns of (-- a, + a)\{o} which preserve Xl{~=o}. If h is a local conjugacy between two saddle-node arcs { %,} and { ~,,}, it must also conjugate the translations defined by {%} with those of{~,}. Let X, X be the smooth vector fields such that X x=q~0 and X t=~0. For x#o, ~-4=o and z=X~(x) near x, let x' = h(x), ~' = h(~), z' = h(z) and z' = Xv(x' ). Since h conjugates T~, and Ti,,,, and T~(z) = Xt(s Ti,~,(z') = ff, t,(~'), we conclude that there is a (conti- nuous) function t'= a(t) such that hXt h-t= "Xol0" From the group property Xt, o X~, = Xt,+t, we deduce that or(t) is linear in t. Since Xx = ~0, Xx = ~0, we have h o X t o h- t = Xx and hence h o X~ o h- t = X~ for all t. This proves the theorem. Proof of Theorem (3-4). -- From Theorem (3. I) and invariant manifold theory (see Section 2 and [26]), it follows that ~, near (s ~), is locally conjugate with ~ near (o, o) where g(x~,...,x,,~)= x,+~+~,247177 . 20 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 2x The stable manifold W* of 70 is {(xt, ..., x=) : x x < o}. We take the strong stable foliation #" of ~ near (o, o) to consist of the manifolds { (xt, ..., xm, ~) : (xx, ..., xh, [z) = constant}. In a neighborhood U ofo, we give a dynamical characterization of the leaves of ,~** c~ ~': qx, q2 e W' belong to the same leaf of ,~" if and only if there are sequences { [zl}ieN, { ~)~e ~ and { q~}~eN such that -- (6, ~) -+ (qt, o) and (6, ~) "-* (q,, o), -- for each i, {~'~(ql) ' and }~-0 ' are contained in U; 9 ~'~ i 9 ~i " -- lim(~(qt), ~) = lim(~m(~), ~t~) artd this limit is not the saddle-node point. This follows from the special form of ~. Let ~'~' be some strong stable foliation for ~. The local conjugacy between and ~ mentioned before cart be chosen to map o ~'~' to o q~''. This implies that o~"' ~ W' also satisfies the above dynamical characterization. Hence, ~"' t3 W' is unique and is preserved under conjugacies. Theorem (3.4) is proved. Note that a local conjugacy of~0, or ~, does not have to respect the above foliation. This has to do with the fact that the dynamical characterization was only possible by using 9 on a full neighborhood of (o, o) in R '~ � R. 4. The flip Here we consider arcs {q~) of diffeomorphisms on R such that ~0(o) = o, (dq~0)0 = -- I and such that the 3-jet of (~0) ~ at the origin differs from the 3-jet of the identity. Such arcs of diffeomorphisms are obtained by restricting a generic arc of diffeomorphisms, at a flip, to a center manifold. The origin is either a source or a sink of q~0; in the following we shall assume it to be a sink of ~0; the other case then occurs for ~o t and is completely analogous. With a coordinate change of the form = ~'(x, ~) we can put r in the form = - + oct') + oCl l. ') where X is a real function and X(o) = o. We shall say that the flip unfolds genzrically dX if ~-~ (o) 4= o. This is, of course, a generic condition. In higher dimensions, we will dX use this teriidnology if ~-~ (o) 4= o on some center manifold. Similar considerations apply to periodic flips. dX Now we return to the one dimensional case. We shall assume ~-~ (o) > o; the 21 o~ S. NEWHOUSE, J. PALIS AND F. TAKENS other case can be reduced to this by replacing "~ by --~. It is clear that the phase portrait looks as follows: ..,.Icf Fixed points \ ,1/ t Points with period two >t J 17161. I The foUowing theorem can be easily proved using the methods of proof used in Theorem (3.5) in the region {~ <__ o}: Theorem (4-x). -- Any arc {%} of diffeomorphisms on It which is of the form %(x) = -- x + x 3 + ~.(p.).x + o(x 4) + o(1 ~[ .x ~) with Y(o) > o is locally conjugate with = - x + x + v.x. 5. The Hopf point We consider arcs {%} of diffeomorphisms of R ~ such that %(0) = o, (d~0)0 has eigenvalues on the unit circle, but different from + z, and such that the 3-jet of 9o makes the origin an attractor or a repeller. We shall assume it to be an attractor; otherwise we consider 9~-1- Up to a change of coordinates, the origin will be a fixed point of % if I ~[ is small. Let ),(~), X(~) be the eigenvalues of (d%)0. For generic arcs one has [),(~)l~-0 :l: o, and we shall say, in this case, that the Hopf orbit unfolds generically. As for the saddle node and flip, we shall use the same terminology for periodic Hopf points if they unfold generically on center manifolds. In the following, we return to dimension two, and we shall assume that ~ Ix(w)[ _o > o; otherwise we replace by -- ~t. These arcs have been extensively studied; for references see [32]. From the fact that, for tt <__ o, the origin is an attractor of % one concludes: Proposition (5. x ). -- If{%} and { ~,} are arcs of diffeomorphisms of R', satisfying all the above requirements, then there is a continuous one parameter family of homeomorphisms h~, ~ < o, from a neighborhood of o E R" to a neighborhood of o ~ R 2, such that h~, o % = ~, o h~, when- ever defined; i.e. (x, ~) ~ (h~(x), ~) is a local conjugacy between % and ~, on ~ <__ o. 22 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 2 3 A consequence of Proposition (5- I) is that arcs {%} as above are locally stable for ~<o. For ~>o this is not the case. Theorem (5.~). -- Let {%,} be a Coo arc of diffeomorphisms as above. Then in any Coo neighborhood of{%} there is an arc {~} such that the arcs {%}, {~} are not conjugate, even not mildly conjugate. Proof. ~ We can first approximate the arc {%} by an arc {~} such that the eigen- values of (d%) 0 have the form e 2"~ with 0r irrational Then we know [4o] that ~',, is rotationally symmetric in the formal sense. By this we mean that there is a smooth S 1 (= R/2~) action R on R~� R, differentiably conjugate with the usual action R : R,(xl, x2, a) = (xl.cos 0r + x~.sin ~, -- x 1 sin ~ + x~ cos ~, a), such that for each ~, R, o ~ and ~ o P,~ have the same oo-jet in the origin. So with a second perturbation one can find an arc { ~'~,} such that the corresponding map ~' defined by ~'(x, ~) = ~(x) commutes with the S x action R, at least on a small neighborhood of (x ---- o, ~ = o). One knows [32] that ~, for a > o, has an invariant circle, say C,. Because ~' commutes with the smooth S 1 action (for a small), ~ [ C~, is differentiably conjugate to a rotation. Since there is a residual subset of the diffeomorphisms of S x no element of which is conjugate to a rotation, there is finally an approximation ~ of ~ (in the C oo sense) such that ~ also has C~, as invariant circle but such that for some sequence { al}~e i, a~ -+ o as i ---> 0% ~ [ C~, is not conjugate to a rotation. It follows from the construction that { ~,} and { ~} are not conjugate and even not mildly conjugate near (o, o). Hence % cannot be (mildly) conjugate to both {r and { ~}. This proves the theorem. Remark (5-3)- -- If{ %} is an arc of diffeomorphisms on a manifold M of dimension m > 2 which has a Hopf point at (~, ~) then the conclusion of Theorem (5.2) remains valid. In the construction of the C ~~ arcs { ~-~}, {qS~}, one has to add in this case a preli- minary step, namely one has to modify { %} first so that it has a Coo centermanifold. Then the modifications, as described in the proof of (5.2) are carried out in that center manifold and extended to a neighborhood. 6. Quasi-transversal intersections We consider a C ~~ arc of diffeomorphisms {%: M-+M} and assume that for some ~ e R and some ~ e M, there is a non-transversal intersection at ~- of a stable and an unstable manifold (of some periodic points) of q~g. In such a situation, if we denote the stable, resp. unstable, manifold by W', W ~, there is a canonical quadratic map D : T~(W') c~ T~(W") -+ T~(M)/T;(W') + T~(W"), which is analogous to the intrinsic 2 nd order derivative [3] and is defined as follows. Let i: T~(W ~) t3 T~(W") -+W ~ be a smooth map such that i(o) = ~ and (di)o is the canonical injection; let z~ be a projection (i.e. ~ = ~:) of a neighborhood U of ~- 23 S. NEWHOUSE, J. PALIS AND F. TAKENS to a submanifold whose dimension equals the dimension of T~(M)/(T~(W') + T~(W")), and such that 7r(~) = ~-, (dTr)~(T~(W 8) + T~(W")) = o and 7r(W*) = k-; let N = re(U). Then T~(N) is canonically isomorphic with T~(M)/(T~(W') + T~(W")). The map 7r o i maps o to ~-, d(~ o i)0 = o; hence the second derivative dg(~ o i)0 : T~(W') c~ T~(W") -+ Ti(N) is a weU defined quadratic map; we define D to be the composition of d~(Tr o i)0 with the canonical isomorphism T~(N) ~ T~(M)/(T~(W') + T~(W")). The maps D does not depend on the various choices. It is not hard to show that, for generic arcs of diffeomorphisms {q~}, all the non- transversal intersection points ~ofstable and unstable manifolds of ~, for ~ E R, s ~ M, will satisfy (in the above terminology): a) dim T~(M)/(T~(W 8) + Ti(W")) = I ; b) D is non-degenerate. Under these circumstances there are coordinates xx, ..., x,, on a neighborhood of x- (~ corresponding to o) such that W' and W ~ locally have the following Canonical form (6..) W 8 = (xx ..... x~_, = o} W" = (x.+~ ..... x,. = o, xl =f(x,._,+...., x.+d} where m = dim(M); s = dim(W'); u ~- dim(W") ; f is a homogeneous non-degenerate quadratic function; if m--s+ x>u+ I then one should read x 1=o. Before proving (6. i), we note the following c) max(s, u) < m by condition (a) above; el) the vector is not in T~(W') + T?(W"); 0 0 , .. are in T~(W') c~ T~(W"). e) the vectors ~x,,_-,+l ~ "' ~x~-+l we can choose a Proof of (6. i). -- Since dim(T~(W ~ + T~(W")) = m -- r, From this it clearly (u + 1)-manifold W" containing W" which is at ~- transversal to W'. follows that one can choose coordinates ~-1, ..., ~,, such that w' = (~1 ..... ~_, = o}; ~" = (~.+2 ..... ~. = o}; W" is tangent at ~- to {xx = x-.+~ = x-.+ s ..... ~-,. = o}. 24 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS ~5 Then, for some functionf w" = {x-,+~ ..... ~ = o, ~ =f(~,,..., ~+d}, f(o)=o and (df)o=o. Now we replace ~t by the new coordinate Xl = "~1 --f(x-9., 9 ..1~ g'-u+l) +f(o, ..., O, X-m--,+l, "'', X-u+l); then we have W m ~ {x 1 ~ ~9. ..... X-m--s ~ O}~ W ~ = {xl =f(o, ..., o, ~-m-,+ 1, -.., x~+l), ~+~ ..... ~m = o}, the first derivative of (x-,~-~+l, ..., x,+l) ~f(o, ..., o, ~,~-,+1, ..-, x-~+l) is zero and the second derivative is just D and hence has maximal rank. Now we can apply Morse's Lemma [3] and obtain a coordinate change of the form xj=~ for 2<j<m--s or u+2<j<m, x~ -= xj(~m_,+ l, . . ., -~+ x) for m -- s + i < j <_ u + I, for which we have W' = {x~ ..... x,,_, = o}, W" = {x~ =f(x,,_,+l, ..., x,+l), x,+, ..... x~ = o), with f homogeneous, quadratic and non-degenerate. Remark (6.2). -- In suitable coordinates the generic unfolding of the quasi- transversal intersection puts the manifolds W~, and W~, in the form W~ = {xl = o ..... x~_, = o), w~ = (x,+, ..... x,, = o, x~ =f(x,,_,+~, ..., x,+d + (~ - ~)). 4 III. -- NECESSARY CONDITIONS FOR STABILITY OF ARCS In this chapter we shall obtain necessary conditions for the various kinds of stabilities of arcs we have defined. Let I = [% x] and let t~ be the space of C | arcs q~ : I -+ Diff~~ with the G ~176 topology. Elements of~ will be denoted by ?, or by { %) when we wish to make the dependence on the parameter t~ ~ I explicit. Recall that ,~ C ~ is the subset of those arcs ? ~ ~ such that (I) '~o E MS, (2) b = b(~) = inf{~ EI : ~. CMS}< i, (3) the limit set of ?~ consists of finitely many orbits. A diffeomorphism f is called elementary if either (a) there is exactly one quasi-hyperbolic periodic orbit, the other periodic orbits are hyperbolic, and all stable, strong stable, unstable, and strong unstable manifolds meet transversally; or (b) all periodic orbits are hyperbolic, there is one quasi-transversal orbit of intersections of stable and unstable manifolds, and all other stable and unstable manifold inter- sections are transverse. Let ~ C ~ be the set of arcs ~p ~ M such that % is elementary and the quasi- hyperbolic periodic orbit of ~, if it exists, unfolds generically. We proceed to discuss necessary conditions for stability of arcs in ~. x. The modulus condition (quasi-transversal intersection) In this section we shall show that left stability of an arc { %} in ~ forces all stable and unstable manifolds of periodic points of ~b to meet transversaUy. Before this, we consider the effect of a quasi-transversal orbit on topological conjugacy. The next theorem shows that generally such an orbit yields at least a one-parameter family of distinct topological types. We will refer to this phenomenon by saying that moduli occur. It should be noted that this occurs even in a locally isolated codimension one submanifold of the boundary of MS on the 2-sphere (hence, on any manifold of dimension larger than one). 2fi BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS For example, consider an MS diffcomorphism % on S z as in the next figure. 2, /\ ~> /\ 0 < 0 > < \/P2 ];'10. 2 The circles represent sources and sinks and there are two saddle fixed points Px and ps. We choose a curve of C s diffeomorphisms {%}, o <~ tL _~ t, starting at % so that px andpl remain fixed for each %, and W"(pt, ?~t) has a single orbit r of quasi- transversal intcrsectiom with W~ 9~) as in the next figure. I/2 (X) o~) 2 fx) ~. (~ ~ / P2 / \ / Fro. 3 This can be done so that % is in MS for ~ 4= - and any perturbation {q~} of{%} has a unique bifurcation b(9' ) near -. Let f be a diffeomorphism of M with a hyperbolic fixed point p. Let =" be the largest modulus of the eigenvalues of dr(p) which are inside the unit circle, and let 8" be the smallest modulus of the eigenvalues of dr(p) which are outside the unit circle. If there is an eigenvalue = of dr(p) such that (x) lml = ~', (2) ~ has multiplicity one, (3) any eigenvalue X ofdf(~) different from ~ or the complex conjugate i satisfies [ ), [ 4= ~*, then we say that the weakest contracting eigenvalue = off at p is defined. Similarly, 27 28 S. NEWHOUSE, J. PALIS AND F. TAKENS if there is an eigenvalue [~ of multiplicity one of dr(p) such that ] ~ [ = [~* and any eigen- value X 4: ~, [~ satisfies [ X 1 4: [~*, we say the weakest expanding eigenvalue [~ off at p is defined. Let p and q be hyperbolic fixed points off such that the weakest contracting eigen- value ~ at p and the weakest expanding eigenvalue ~ at q are defined. Let t-I v be a C 1 invariant manifold containing W"(p) and tangent at p to the sum of the eigenspace of ~ and TvW"(p), and let I-Iq be a C t invariant manifold containing W'(q) and tangent at q to the sum of the eigenspace of ~ and TqW'(q). The existence of I-I v and Hq is.proved in [I2]. They are not unique. However, the tangent bundle of Hp along W"(p) is unique, and so is that of Hq along W'(p). Let r be a quasi-transversal intersection of W"(p) and W'(q). We say that r is a regular quasi-transversal intersection if W"(p) is transverse to Hq at r and W~ is transverse to Hp at r. This definition is independent of the choice of the manifolds H~ and Q.q because it depends only on THp, THq along W"(p), W~ respectively. Note that part of the definition of a regular quasi-transversal intersection of W"(p) and WS(q) is that the weakest contracting and expanding eigenvalues ~ at p and [~ at q be defined. Note that for arcs q~ in a residual subset of ~ all quasi-transversal orbits of %, are regular. Theorem (t. i ). -- Let f, f' be C 2 diffeomorphisms. Let p and q be hyperbolic fixed points having an orbit {f~(r)} of regular quasi-transversal intersections of W"(p) and W'(q). Let o~ be the weakest contracting eigenvalue off at p and ~ be the weakest expanding eigenvalue of fat q. Make analogous assumptions on f' concerning fixed points p', q', eigenvalues ~', ~', etc. If there is a conjugacy from f to f' defined on a neighborhood of the closure 0f{fi(r) } mapping p to p', q to q', and r to r', then log 10: I log 1~'1 log log I '1" Remark x. -- Ifp and q are periodic points of period x(p) and ~(q) instead of fixed points, and n is the least common multiple of v(p) and x(q), the theorem can be applied to f". Doing this one obtains that if ~ is the weakest contracting eigenvalue off "Ip/at p and [~ is the weakest expanding eigenvalue off "cq~ at q and 0(, ~' are the corresponding eigenvalues for f', then the existence of a conjugacy between f and f' implies log I~ I log I~'l logl l Iogl 'l 2. Theorem (I. I) shows that quasi-transversal orbits lead to at least one dimen- sional invariants of topological cortjugacy. It is interesting to ask what additional invariants in the presence of such orbits are sufficient to imply the existence of a topo- logical conjugacy in various contexts. For some results in this direction, see [I8], [28]. 28 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS Proof of Theorem ( x . i ). A) The case m=2; re=dim(M) In this case the dimensions of the stable and unstable manifolds are both one. Replacing f, f' by f~, f,2, if necessary, we assume ~, 6, ~', 6' > o. So we are in the following situation: __ W u (p~ W s ~ql \/ /\ FIG. 4 We consider a sequence of points rl converging to r but so that r~ r (W"(p) u WS(q)) for all i. By choosing a subsequence if necessary, we can arrange that there are sequences of integers n~ ~ oo, ml -+ oo with the property that f-'~(r~), f'~(ri) , has a limit in WS(p) -- p, W"(q) -- q, respectively. Let p(ri, W"(p)), p(r~, WS(q)) denote the distance from r i to W"(p), respectively W"(q), with respect to some Riemannian metric. Because f is C 9", it is C 1 linearizable on WS(p) and W"(q) [8]. From this, we conclude that p(rl, W"(p)) ~ e'~ and p(ri, W"(q)) ~ 6 -~, where ~ denotes equality up to a positive multiplicative factor, depending on i but uniformly bounded and bounded away from zero. It is clear from the picture, or rather from the normal forms in Chapter II, that a sequence ~ can be chosen so that p(rl, W"(p)) ~ p(r~, W"(q)). In that case we have log 0~ _ lira mi log 6 i-~| n~" Now we assume that there is a local conjugacy h, defined on a neighborhood of the closure of the orbit of r, as in the theorem. Let h(r~) ----- r'~. From the topology of the intersection of W ~ and W * and the position of the r~'s (see figure) it follows that p(r~, W"(p')) < p(r;, W'(q')); i.e. p(r , w"(F)) < i. p(r , w'(r - 29 3o S. NEWHOUSE, J. PALIS AND F. TAKENS Since (f')-'~(r~) and (f')"~(r~) must have a limit in W'(p') --{p'} resp. W"(r --{r wc conclude that p(r~, W*(p')) ,-~ (Qt')~ and p(r~, W~ ,~ (f~')-=,. This, together with p(r , < i pCr , w'(()) - implies that log oc' m, log ~ log oc' > log <~ -- lim -- = --. Reversing the argument, one finds log ~' -- log log ~' -- ~-,~o n~ log and hence log ~' log 0~ log ~' = log ~" Observe that if W"(p) is transverse to W~162 and W"(p') is transverse to W'(q'), then we cannot conclude for any sequence r~ -+ r that I~(r~, W~(P')) is bounded. So p(r , w'(q')) we needed some of the properties of the " topology of the intersection ". In the next higher dimensional case we have to analyze this in detail (1). B) The case dim(W"(p)) = dim(W"(r = m -- x As in case A we assume a, ~, 0~', ~' > o and we consider sequences ri --* r such thatf-~(rl) converges to a point in W'(p) --{p}, and f~(ri) converges to a point in W"(q) -- {q} for some sequences nl, m~ -~ co. We also assume that one of the following three possibilities takes place (this can be obtained by taking a subsequence): p(r. W"(p)) is bounded and bounded away from zero (or ,~ x); p(r,, w'(q)) --, co. In these cases we find that < lim inf(~) i.-* oo -- ; log fi ~-'| -- ; > lira sup -- . -- i--*. eo We assume that there is a local conjugacy h and denote h(r~) by r~. We assume also that p(r~, W"(p')) either goes to zero, is ~-, I, or goes to co; in each of these cases we get w'(q')) log ' tim" a relation between ~ and " -- as above. (~) After the arguments for sections B, C, and D were written, a cleaner treatment was discovered by S. VAs STRreN. This treatment uses arguments pre~ented in the proof of Theorem (2.1) in lisa]. 30 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS log ~ < log ~' From this we claim that, if ~ log ~" then, for each sequence r~ -+ r as above, pCr,. w"(l,)) p(r., w"(l,')) ~- o and/or ~. oo. 0(r,, w'(q)) p(~;, w'(q')) log ~ log ~' Indeed suppose that ~ < log ~' and r~ -~ r as above. If does not go to 0% then for some constant 0 > o, we have (~')'~(~')'~ <_ c. This gives n~ log ~' + m~ log ~' <__ log C or log ~' log G m~ log ~' -- n~ log ~' hi" If 8> z is so that log0c+log8 log~' = -- then log ~ log ~" log < log C rn~ log ~ -- n~ log ~ n~' or n i log ~8 + m~ log [~ < log C log -- " log [$'" This means that (a~)~, is bounded, so ~'~ ~"~-+ o as i --~ oo which implies that p(r,, w"(p)) 9 ---~ O, p(r,, w'(q)) logoc log:C This proves the claim. Similarly, for ~ > log ~' 9 ,~e have p(~,, w-(p)) p(,;, w-(p')) oo and/or --> o. p(r,, w'(q)) p(~, w'(q')) We want to prove that this cannot happen for every sequence in case of a quasi-transversal intersection of W"(p) and W'(q) (and W"(p') and W'(q')). For this we take subsets M, A ~, A ~ of a neighborhood of r such that: -- A' u A " u A ~ is a neighborhood of r; ~(,,, w"(p)) -- if rl-',-r is a sequence such that -+o, then r~A ~ for i big; p(r,, W'(q)) if r i ~ r is a sequence such that p(r~, W"(p)) ~.o% then r~A ~ foribig; p(r,, W'(q)) r~ r WU(p) t3 W'(q)) then --if r i-)-r isa sequence such that r~eA ~ (and p(n, w-(p)) e'~a I ~ p(r,, w'(q)) a_____~ W'(q'))W*(P')) p(r~,P(r~' s. NEWHOUSE, J. PALIS AND F. TAKENS A TM, A" and A '~ are similar sets in a neighborhood of r'. From the above arguments _ log ~' it follows that if log 0t < then there is a neighborhood U of r such that log ~ log [3" h(U n (h ~ u A')) C A" log ,t log a' (and if ~>log~,, h(U n (A ~ w A")) C A'"). In order to derive a contradiction from this (namely from the assumption that log a< log 0d I in the case of quasi-transversal intersection, we proceed to an explicit log ~ log ~'] construction of A ~, A' and A ~ From Chapter II we know that there are coordinates xl, ..., x,~ in a neighborhood of r such that locally W'(q) = {x. = o} and W"(p) = {x m = ~(xl, ..., xm_x)} where ~ is a homogeneous quadratic function. In these coordinates we take { , } A': Ixml <_ g I ~(xx, 9 9 ~m_x)l A" = Ixm -- ~(Xl, ..., x,~_l)l < g I ~(xl, ..., xm_l)l / ' I A ~ == closure of the complement of A' u A ~. Near r', W~(p ') and W"(q') have the same form, so there is a diffeomorphism ~" from a neighborhood of r to a neighborhood of r', mapping W"(p) to W"(p') and W'(q) to W'(q'). We define A" = LF(A'), A '~ = ~(A ~) and A '~ = ~(A~ It now follows that ~ = ~- 1 o h is a local homeomorphism from a neighborhood ofr toitself, inducing homeomorphisms in W"(p) and W*(q) and mapping (A ~ u A*) n U into A'. Let U*CU be a subset of the form U* ={(xt, ...,x=) [o< Y.~< a*} and U**D~(U*) a set of the form U** ={(xz, ...,x,,) l o< Y~x~< a**}. From the fact that the subsets W~(p), W*(q), A ~, A' and A ~ restricted to U* or U** are all cone- formed, and the fact that ~ is a local homeomorphism inducing local homeomorphisms in W~(p) and W*(q) we conclude that the maps o" n W'(q) -~ u'" n w'(q), o'- (w"(p) n u') -+ o"- (w"(p) n u"), induced by ~, induce isomorphisms ill the homology. From the definitions it is clear that the inclusions U' n W'(q) -+ U* n A', U" n W'(q) ~ U** n A', U" n (A' w A ~ ~ U* -- (W"(q) n U*), U** n (A' w A ~ ~ U** -- (W"(q) n U**), 32 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS are all homology equivalences. Now consider the morphism of long exact sequences: H,(U* n M) ~ Hi(U* c~ (A* u A~ ---> Hi(U* n (A' o A~ U" n A') -~ H,(U*" c~ A') -+ H,(U*" n (M o A~ -+ H,(U" n (A' w A~ U*" n A) hi,/h and ha axe all induced by ~, so hx,/h are isomorphisms. By the Five Lemlna, ha is then also an isomorphism. But since ~(U* c~ (AS u A~ C U** c~ A", ha is the zero morphism. Hence HI(U* c~ (A* u A~ U* c~ A *) = o for all i. This is in contra- diction with the following result, which follows from standard arguments: if ~(xa, ..., x,,_~) = ~ -t- ... + ~-- 4+~--... -- xs,,-t then Z if i=k or i=m--k--I provided k4:m--k--i; H~(U*n(A'uA~ ZSZ if i=k=m--k--I; o otherwise. C) The general case: ~, ~, od, ~' real We assume that f has fixed points p, q and invaxiant manifolds Hp and Hq as in the theorem and ~, [~, 0:', ~'> o. Then, dimHp dimW"(p) + i, and dim Hq = dim W"(q) + i. Let Hp,, Hq, denote the analogous manifolds for f'. Of course, if there is a conjugacy h betweenf andf', it does not follow that h(Hp) = Hp, or h(Hq) = He. If we knew that h(Hp) ---- Hv, and h(Hq) = Hq,, we could apply the argument of case B for sequences r i->r in Hpr~Hq. For in HpnHq, the manifolds WU(p) n Hp (~ Hq and W"(q) n Hp n Hq have codimension one, and intersect " nearly " quasi-transversely. We say nearly because, since Hp and Hq are only C t manifolds, a quasi-transvelse intersection of WU(p) n Hp ~ Hq and W'(q) n Hp n Hq is not defined. However, we Call choose C 2 submanifolds Hp and Hq (not necessarily invaxiant) which are G I close to Hp and Hq, respectively, such that Hp n ~p D WU(p), Hq c~ Hq D W*(q), W"(p) n H, n Hq = W"(p) n H, n Hq, W"(q) n Hp n Hq = W"(q) r~ Hp n Hq, and W*(q) c~ Hp n ~q intersects W"(p) n Hp n Hq quasi-transversely. Hence, the metric properties of the intersection of WU(p) c~ Hp r~ Hq and W*(q) c~ Hp r~ Hq in Hp c~ Hq are those of a quasi-transversal intersection. Since h does not map Hp to Hp, or Hq to He, we must modify our arguments. We shall show that there is a map  from h(Hp n Hq) to H, n He such that for any 5 S. NEWHOUSE, J. PALLS AND F. TAKENS x 9 h(Hp n Hq) -- (W"(p') u W'(q')), we have p(x, W"(p')) ,-~ p(~(x), W"(p')) and p(x, w'(r w'(r Let N be a small disk neighborhood ofp where we can define a C 2 coordinate chart such that the components of WS(p) n N and W"(p) n N containing p are coordinate planes. In the sequel, we will xestrict ourselves to such a neighborhood N and to the components of W"(p) and W~ containing p. Let % : N -+ W'(/~) be the natural projection and assume that the negative orbit of r lies in N. Since W'(q) is transverse to Hp at r, we can characterize W~ as the subset of W~ where the critical points of ~, ] W'(q) accumulate. This characterization is based in the following fact: If v 9 W~ then v r W"(p) if and only if there are a neighborhood V of v and a neighborhood U of r such that for each v' 9 V n W~ there is an arbitrarily small neighborhood V' of v' in V such that, for n 9 N suficiently big, {f-"(W~ n U) n V'} C{f-"(W~ n U) n V} is a homotopy equivalence. In this formula we should omit from f-"(W'(q) n U) those points whose orbits from W~ n U to f-"(W~ leave N. From this dynamical characterization it follows that h(W"(p))= W~ ') and similarly h(w""(q)) = w.-(q'). Thus h(Hp) n W'(p') is an invariant C o curve off' which meets W"(p') only at p'. Sincef' is C t linearizable on W~ ') [8], we see that h(Hp) n W~ ') has to lie in a small sector about Hp, n W'(p ') which has width zero at Hp, n W"(p') as in the following figure. >> << W `s (p') J L Hp, n W s (p') Fm. 5 34 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS From this it follows that for any smooth projection ofa neighbourhood of Hf to Hv, , the distance of points in h(Hp,) to W"(p ') is not changed essentially, i.e. the quotient of the new and old distances is uniformly bounded away flora zero and infinity. It is not hard to choose such a projection al on Hf which does not essentially change (at least near r') the distances to W'(q'). Using the same type of arguments one can define a smooth local projection a, on H e, in such a way that -- ~ maps Hp, into itself -- ~, does not essentially change distances from W~(p'). As before, it follows that ~2 does not essentially change the distances from points in h(Hq) to W~(q'). So ~2o~1[ h(HpnHe) :h(HpnHe) -+Hp, c~H e, has the required properties. To see that the whole reasoning of case B applies, we first choose metrics on Hp, H, H ,, H e, induced by nearby C 2 manifolds Hp, Hq, Hf, He', respectively. Deft- rung A', A ~ A" C Hp c~ He as before and projecting them into Hp and H e to obtain A 8, A ~ A", we then have that the maps W"(p) n Hp n H e -+ A ~, W'(q) c~ Hp c~ Hq -+ A', etc., are homotopy equivalences. Now the argument can be completed by observing that (I) (~ roughly preserves distances; (2) a induces, locally at r', homotopy equivalences from W'(q') n h(Hp n I-Iq) to W'(q') c~ (Hp, n He, ) and from h(Hp n Hq) -- W"(p') to (Hf n He, ) -- W"(p'); (3) if r~-+ r', r~ e h(Hp n He) are such that f-'~'(r'), respectively aOm(r~), has a limit in WS(p ') --p', respectively WU(q ') --q', then o(r[,W~(p')) ,~, (0d)m and v(r;, W'(q')) These facts follow easily from the constructions. D) The general case: ~, od are not real or ~, ~' are not real Let Q be the df(p)-invariant subspace of TpW'(p) complementary to the eigenspace of 0c, and let Q be the df(q)-invariant subspace of TqW~(q) complementary to the eigenspace of ~. Let W'8(p) be the invariant manifold in Ws(p) tangent at p to Q., and let W*"(q) be the invariant manifold in W"(q) tangent at q to Q. If Hp and Hq are the invariant manifolds in the statement of Theorem (I. I), then W**(p) is transverse to Hp at p and W~"(q) is transverse to Hq at q. The fact that W"(p) is transverse to H e at r enables us to get a dynamical characterization of W""(q). Similarly, we will obtain a dynamical characterization of W"(p). Once these are obtained, it will follow that a conjugacy h from f to f' as in the statement of Theorem (i.i) will have the property that h(W'(p)) = Ws'(p ') and h(W""(q)) = W""(q'). From this, it will follow as in case C that h(H e n W'(p)) is in a small sector about Hp, t~ W"(p') which meets W"(p') only at p'. Similarly, h(H e c~ W"(q)) is in a small sector about H e, n W"(q ') which meets W'"(q ') only at q'. The arguments then proceed as in case C. In the present 35 S. NEWHOUSE, J. PALIS AND F. TAKENS situation either or both of W~(p) and W'(q) will have codimension 2 in Hp and Hq, respectively. This changes the arguments slightly, but, since these changes are straight- forward, they will be left to the reader. Thus to complete the proof of Theorem (x. I), we must show how W~"(q) is characterized dynamically. Let U be a small disk neighborhood of q containing the forward orbit of r. We can take U as a coordinate chart with a Cl ~ coordinate system having the components of W~(q) and W*(q) containing q as coordinate planes. Let ~ : U -+ W~(q) be the natural projection. For an integer n> o let ft~ denote fn] N f-J(U). Define 0<j<n W~(q) to be the set of points y ~ W"(q) n U such that (.) for each sequence o < nx< na< ..., the critical points of ~, ]f~(W"(p)) accumulate on y as i ~ oo; in case dim(W"(p)) + dim(WS(q)) = dim(M) -- I), f~(W"(p)) accumulates on y. We assert that W~"(q) =U ~>0f'( w" to~(q)). This implies a dynamical characte- rization of W"(q). To prove this assertion, first suppose that flU is linear and, near r, the set of critical points (or fold points) L of ~, ] W"(p) is an affine subspace. Here we think of U as an open subset of Euclidean space via linearizing coordinates. Note that dim(L) = dim(W"(q)) -- i. If the eigenvalue ~3 is real, the assertion follows as in the previous case when the eigenvalues 0~, [~ were taken to be real. So assume ~ is not real. Assume also that Hq, W"(q) and W"(q) are linear subspaces of U near q. Choose an affine subspace L 1 C L which is complementary to Hq at r. Then f"(L1) converges to as n -+ or. This implies that (.) holds for any y e W~(q) near q. Now let y be in W"(q) -- W~'(q) and near q. Since f] Hq n W"(q) near q is a rotation, there is a sequence of integers n 1 < n~ < ... such that f~i(L n Hq) does not accumulate on ~y as z -+ oo where 7~ : W"(q) ~ W~(q) n Hq is the natural projection. This implies thatJ~(L) does not accumulate ony as v -+ or. Tiffs proves the assertion if f] U is linear and L is affine. The extension to the case wherefis not linear or L is not affine near r is straightforward. We leave the details to the reader. Theorem (i. I) is proved. Lemma (1.2). -- Suppose f is an elementary diffeomorphisra and x is a quasi-tra~ersal intersection of W~(p,f) and W'(q,f) with p and q periodicpoints off. Then f is not topologically conjugate to any Kupka-~Tmale diffeomorphism. Moreover, if f' is also elementary and h is a topological conjugacy between f and f', then h (x) is a quasi-transversal intersection of hW~(p, f) and hW'(q, f). Proof. ~ This follows from the fact that a transversal intersection of two submani- folds W' and W" of a manifold M is topologically different from any quasi-transversal intersection of W' and W ". In a point of quasi-transversal intersection, either dim W' -4- dim W ~ 4= dim M 36 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 37 in which case the intersection is not a manifold of the right dimension or, dim W ~ + dim W ~ = dim M in which case the intersection number is zero. See Chapter II, section 6. Note that not all non-transversal intersections are topologically different from transversal ones. Theorem (*.3)- -- Suppose f, p, q, ~, f~, and r are as in the hypotheses of Theorem (1. x). Then there is a residual set ~t C ~ such that if ~ e ~x, then no %, o <__ ~ <__ i, is topologically conjugate to dr. Proof. ~ There is a residual set 9~0 C 9 ~ such that if $ e 9~0, then (i) each %, is elementary or Kupka-Smale; (2) if some %, has a quasi-transversal orbit, then the hypotheses of Theorem (,. i) with %, =f' are satisfied; (3) there are at most countably many lz's for which %, has a quasi-transversal orbit. Now, since each % has at most one quasi-transversal orbit, it follows that the quotients of the logarithms of the moduli of quasi-transversal orbits occurring in {%} form a set which is at most countable. With standard arguments, one can show that there is a residual subset ~1 of 9~0 such that if 9 z t~x, then all the quotients occurring for ~ are different from log [0c] Now Theorem (,. 3) follows from Theorem (,. x) and Lemma (I. 2). log [ ~ [" Corollary (*.4). -- If ~ e ~ is left stable, then all stable andunstable manifolds of periodic points of 9b intersect transversally. Proof. -- Since q~ e ~, we have that % is elementary. Thus, if % =f has a quasi-transversal orbit, then we may assume the hypotheses of Theorem (i. i) and hence Theorem (i .3) are satisfied. By Theorem (I .3), q~ is not left stable. 2. Necessary conditions for mild stability and stability Suppose q~ e ~ and ~(p) is a saddle-node for %. We say ~(p) is s-critical for % if there is a periodic orbit r such that W~'(~(q)) has a non-transversal intersection with the strong stable foliation o ~-s' of W~(~(p)). Similarly, we say ~(p) is u-critical if it is s-critical for q~-l. If ~(p) is either s-critical or u-critical but not both, we say that ~(p) is semi-critical. If ~(p) is both s-critical and u-critical, we say it is bicritical. Proposition (2. ,). -- If ~ is mildly stable, then it is left stable and the quasi-hyperbolic periodic orbit is not a Hopf orbit or bi.critical saddle-node. 8~ 3 8 S. NEWHOUSE, J. PALIS AND F. TAKENS Proof. -- The first statement is obvious and the second statement was proved in Section 5 of Chapter II. Now suppose r is a bicritical saddle-node for ~b" We assume p is a fixed point of ~b" If qx and q, are periodic points of % such that W"(qx) has a non-transverse intersection with the strong stable foliation of W'(p) and W'(q,) has a non-transverse intersection with the strong unstable foliation of W"(p), then, perturbing if necessary, we may assume that these intersections are quasi-transverse. For some ~ near b, it then follows that W"(qa~, q~) has a quasi-transverse intersection with W'(q~,, q~). Here, of course, qx~ and q~ denote the unique hyperbolic periodic points of ~ near ql and qz, respectively. Using Theorem (1.3) we can perturb to ~' in ~ so that no q~', is conjugate to %,. This shows that ~ is not mildly stable. Proposition (2.2). -- If 9 is mildly stable, then ~b has no cycle of length bigger than one. Proof. -- Suppose q~b has a cycle of length bigger than one. We show ~ is not mildly stable. Since q~ ~ ~, q~b has a saddle-node orbit *(Po) which is contained in every cycle. Indeed, the transversality of the stable and unstable manifolds implies that if 9b had any cycle not containing a saddle nodes, then ~b would have transversal homo- clinic points. This would give 90 infinitely many periodic points [34]. Let r 9 9 -, r be the distinct orbits in the cycles of q~b- Replacing { q~ } by some power { ~, }, we assume that all the p+'s are fixed points. Let W"(P0), resp. W"(p0) , denote the strong stable, resp. unstable, manifold of Po. Since L(~b) has finitely many orbits it follows that W"(p~) n W'8(po) = o and W'(p+) n W"(p0) = o for 0 < i < r. Otherwise, q~b would again have transversal homoclinic points. By transversality, we have dim W"(p+) = dim W"(po) or dim W"(p+) = dim W"(po) -- i for each o < i < r. Replacing {~} by {~-1}, if necessary, it is enough to consider the case in which there is a I <j < r such that dimW"(p~) = dimW"(Po). This implies dim W*(pj) n W"(Po) = o and hence Po is u-critical. Observe that, having dim W"(pj) = dim W"(P0) for some i < j < r, we may indeed assume that for all o < i < r (a) dim W~(p+) = dim W"(Po) , (b) W~'(p+) is transverse to the strong stable foliation ~-" of W'(Po), (c) e w"(p+) w"(po) = {Po}. For suppose (a) or (b) failed for some I < i < r. Then Po is s-critical. Since dim W"(pj) = dim W"(Po) , we already know that Po is u-critical. Thus Po is bicritical and Proposition (2. I) has already ruled this out. Let us now suppose that (b) fails for i = o; that is, W"(P0) has a nontrartsverse intersection with .~'". We already know that W"(p~) is transverse to ~" for I < i < r. Perturbing % if necessary, we may assume that W"(p0) has a quasi-transversal intersection with some leaf F in ~". Choose disks Dx C W"(P0) and D 2 C F so that D x and D2 have a quasi-transversal intersection. Let 38 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS be an invariant center manifold in W'(P0). Since W"(pt, b) meets W'(po) transversally. and is transverse to o~', it follows that each component of W~(px, b) n W'(P0) is an embedded curve whose closure near P0 can be given as the graph of a smooth function from ~ to W"(po) as in the next figure. n WS (Po) ~ ~~ Fro. 6 As ~t increases, pieces of W"(Pl, ~t) will sweep across the local part of W'~o ) near Po, so for each Ez > b near b, W"(px, ~t) contains a disk Dx,~G ~ near D1. On the other hand, W'~x , b) accumulates backward on W"(p0). For certain tz's near b and greater than b, W'(pt, bt) will contain a disk D2,~C r near D,. Considering the continuous movements of Dx,~ and D2.~, as ~ varies, one sees that for certain ~t's near b and greater than b, Dx,~ has a quasi-transverse intersection with D~,~. Thus, if (b) fails for i = o, there are ~t's near b for which W"(pl,~) has a quasi-transverse intersection with W'(Pt,L,)- In view of Theorem (x-3), we conclude that ? is not mildly stable. Finally if (c) failed for some i, then d W"(p~) would meet a fundamental domain for W'(p0). Since all intersections of W"(A) and W'(pk) are transverse, this implies W"(p~) n W"(p0) . ~ as in the Morse-Smale case ([24]; Lemma (i.5)). As we have already mentioned, this gives transverse homoclinic points--an impossiblity. Thus, we assume (a), (b), and (c) hold for all o < i < r. We proceed to derive a contradiction from this, and, then, Proposition (2.2) will be proved. From (a), (b), and (c) it follows that each component of W"(pi) n W'(P0), o < i < r, is an embedded curve whose closure near P0 is the graph of a smooth function from the center manifold ~ to W"(p0). Let Pi, be a fixed point such that W~(p~.) n W'(P0)  ~, W"(Po) n W'(A.) 4= o, and for each x <i<r with i+io, W"(p~.) nW'(A ) =o. Then, W"(pl.) nW'(P0) consists of finitely many curves which are permuted by ~Pb. Also, the boundary of each suchcurveis{po,p~,}. By the boundary of a curve-(, we meand~, -- y, ofcourse. Since W"(p~.) is transverse to W'(po) , we conclude that the closure of W'(po) near Pi. is a finite union of closed half-spaces bounded by W'(p~.). Also, since W"(po) has a transverse intersection with W'(pi, ) and dim W"(po) = dim W"(p~,), W"(po) accumulates on W"(pl,) in the G I sense. Thus, we see that W"(P0) n W'(po) has a component T which WU(pl) 40 S. NEWHOUSE, J. PALIS AND F. TAKENS is an embedded curve whose boundary is {P0, x} where x e Ws(p~.) -- {pl.} n W"(Po). This is depicted in the next figure. Fxo. 7 We will show that the existence of T leads to a contradiction. Let D be a small half- disk neighborhood about Po in W"(Po) so that %(OD -- W"(po) ) n (0D -- W~(po)) ---- o. Let D" = cg(~bD -- D) be a fundamental domain for W"(p0). Choose N > o so that n ~ N implies that ~;-"x ~ D. It follows that, for n > N, %-"? is an arc whose intersection with D" -- W"(p0) has at least one component joining the two components of O(D" -- W~'(p0)). We may suppose that 0D" is transversal to each curve ~gy, r E Z. Hence, the set E of points y ~ D" -- W~(p0), for which there is a sequence Yl ~ ?;-,i(y), ni ~ oo, such that yi -+y, is uncountable. But E Cct Ws(p0) n D". We assert that E n W'(P0) ----- o. For the moment, assume this assertion holds. Then E C [.J W'(p~). Since dim W'(pl) + dim W"(P0) = dim M, this would put E a<i_<r in the countable set D ~ n [3 WS(pi) ) which is a contradiction. l<i<r Thus, we must prove that V, n W'(po) ---- o. Suppose E n W'(po) ~ o and let y ~ E n W'(P0). Choose y~ E q~b-"i(T), ni -+ 0% so that yi --~.y as i -+ oo. We claim that yi goes to infinity in WS(po) ; i.e. given any closed disc F C W'(P0) transverse to D" there is an i o~'o so that .y~r for i>i 0. This follows from the facts that 9~'~, n 9;V---- ~ for m~e r and F n D" has only finitely many components. Now let D' and Ds' be fundamental domains for W'(p0) and W"(po), respectively, and let D "~ be a fundamental domain for W~'(po). There is an N ~> o so that ~?~(y) e D ". Since 9~(y~) converges to the point ~(y) in D s and %s(y~) goes to infinity in WS(po), there is an m~  o such that ?~+mi(yi) accumulates on D ~". Thus, d W'(po) n D ~" ~e o. But this implies that % has transversal homoclinic points which is impossible. Thus E n W'(P0) ---- o and Proposition (2.2) is proved. Proposition (~. 3). ~ If ~ is mildly stable, then ~b cannot have a non-critical i-cycle. Proof. ~ As before, if % has such a cycle, it must have a saddle-node ~(p) which is in the cycle. Since the cycle is non-critical, W*(r is transverse to the strong stable 40 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 4 z foliation of W'(r(p)) and W*(o(p)) is transverse to the strong unstable foliation of W"(,(p)). Let nx be the period ofp. Using invariant manifold theory one sees that W"(~(p)) n W'(~(p)) is a finite union of C ~ circles Slb, ..., Skb which are permuted by 9~ '. Taking n i larger, we may assume ~b (Sib) S~ for I < i < k. Replacing 9 by a suitable C ~176 approximation, we may assume for ~ near b that there is a C ~~ circle S~g near Slb which is invariant by q~', and the rotation number of 9~ ' [ S~ is not constant as a function of ~ near b. From the recent work of M. Herman [9] we may find a near b such that 9~,'[ Sl~ is C oo conjugate to a geometric rotation through an angle with 0~12~ irrational. By a small modification of r to q~', we may make ~p~"' [ Sl~C ~~ conjugate to a rotation through an angle a' with 0c'/2~ rational. This means that %, has uncountably many periodic points of the same period. But, since each r is elementary, it has only finitely many periodic points of a given period. Thus ~ is not mildly stable. Remark. -- The previous propositions leave open the possibility that a mildly stable arc might have a critical saddle-node in a t-cycle at %. We will wove in section 4 that this cannot occur if the stable or unstable manifold of the saddle-node is one dimensional. In particular, it cannot occur if dim M = ~. In general, for dim M > 2, we feel that this cannot occur, but we have no complete woof at this time. The next proposition shows that the situation regarding stability is better. Proposition (2.4). -- If q~ is stable, then 9b has no cycles and no semi-critical saddle-nodes. Proof. -- We have already taken care of cycles of length bigger than one and non-critical i-cycles in Propositions (2.2) and (2.3). Suppose ~b has a semi-critical z-cycle containing the saddle-node orbit ~(p). We assume p is fixed by % and W"(p) has a non-transverse intersection with the strong stable foliation .~* of W'(p). The other cases are handled similarly. Perturbing, if necessary, we may assume that o ~'~' is a C 2 foliation, that all intersections of W"(p) with leaves of o ~-"* are transverse or quasi- transverse, and that all the eigenvalues of d% on T~W'(p) have multiplicity one with distinct norms. Let 0c be the weakest contracting eigenvalue of d% on TvW*'(p), and let Cp be a C t invariant manifold in W"(p) tangent at p to the eigenspace of a. Let Hp be a C t invariant manifold containing W"(p) and Cp as in the woof of Theorem (t. i). Let x e W"(p) n W'(p) be a quasi-transversal intersection of W"(p) and ~-~' at x. If y is a curve in Hp n W*(p) transverse to W"(p) at x, then Y is transverse to o~-~' in WS(p) at x and, hence, projects diffeomorphicaUy near x along the leaves of o~-" into an invariant center manifold ~ for %. Let D be a fundamental domain for W"(p). Thus D is a compact set in W"(p) such that 9~D nD = o and W"(p) --{p}C [.J ~](D). Let n~Z U be a small compact neighborhood of D. If y e 7 is near x, then O(Y, x) ,~ [ ~ I "Iv) where n(y) is the least positive integer n such that q~-"(y) ~ U. Here, as in the proof ofTheorem (z. t), we use p(y, x) -~ [a[" to mean that ~ is bounded and bounded away from zero independent of n. 41 S. NEWHOUSE, J. PALIS AND F. TAKENS 4 ~ Now let 9' be a perturbation of 9 such that b(q~') ---- b(~), [a'[ < [0~l, and ~'~ is still a C 2 foliation. Using Chapter II, we may assume that on the center manifold through p, 9b is the time-one map of a C | vector field X b which vanishes only at p. Also, there is a corresponding vector field X; for ~. Suppose there is a continuous conjugacy { h~ } between q~ and ~'. By Theorems (3.2) and (3.4) of Chapter II, we know that (I) h b is C ~~ along ~--{p}; (~) h b maps X b to X;; and (3) hb maps the strong stable foliation ~'*' to the corresponding strong stable folia- tion o ~'u. Since ~'"' is C 2 near ~", all intersections of W"(p') and leaves of ~'"8 will be transversal or quasi-transversal. As in the proof of Lemma (i. 2), this implies that hbX is a quasi-transversal intersection of W"(p ') and o~-;,b,. '" Applying the reasoning in section D of the proof of Theorem (i. i) to W~"(p), we see that, since H r is transverse to ~'~', the invariant manifold in W88(p) tangent to the sum of the eigenspaces complementary to the eigenspace of 0r has a dynamical characterization. It is the set of points where the backward orbit of~'~ ' accumulates. Thus, hb(H p n W"(p)) is in a sector about H~ c~ W"*(p ') in W'8(p'). This implies that ify is near x in y and p(x,y)--~[a[ "lu), then p(h~, hbX ) ~ 10r Hence, (4) lim p(hby' hnx) -- o. v-~ p(y, x) If r~ : W*(p) ~ ~ is the projection along the leaves of ~'b", then (4) implies that hb has derivative zero at ~(x). But this contradicts the fact that hb(Xb(~x)) ---- X'b(hb~(X)). The proof of the fact that stability of ~ implies that % has no saddle-node orbit r which is semi-critical via some other periodic orbit r is similar. If say W*(r has a non-transverse intersection with o~'~', we repeat the preceding argument replacing WS*(r by WS(r The remaining case of W*(~(q)) having a non-transverse inter- section with ~'~" follows replacing { %} by { ~- 1}. Remark. -- We observe that stable arcs in d must lie in ~. This follows from a somewhat more general observation, namely that any stable arc {q~}, not necessarily starting in MS, with a bifurcation for bt = b such that the limit set of ~b has finitely many orbits has the following properties: -- all stable, strong stable, unstable, and strong unstable manifolds of ~b intersect transversally; -- % has exactly one non-hyperbolic periodic orbit, which is either a flip or a non-critical saddle-node without a cycle; this non-hyperbolic periodic orbit unfolds generically; such a ~b can have no cycles because it has finitely many orbits in its limit set; -- there is an r such that if ~ c (b -- r b q- r -- {b}, then %, is MS. 42 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS This statement follows from the previous arguments in the following way. If all the periodic orbits of % are hyperbofic then, since the limit set has only finitely many orbits and % is not structurally stable, there must be a non-transversal intersection of a stable and an unstable manifold. By the results of section one of this chapter the arc { %,} would then be unstable. Also, the non-hyperbolic periodic points have to unfold generically otherwise we would not even have local stability. It is also clear that no more than one orbit of % can be non-hyperbolic. If % has a Hopf point or a non-critical I-cycle, then there are nearby arcs with a smooth invariant circle with irrational rotation; as observed in the present section that also contradicts stability of the arc. Also in this section we saw that a saddle-node with criticaUity and or a cycle of length bigger than one is impossible if the arc { %,} is stable. If ~b has a non-transverse intersection of the stable, strong stable, unstable, or strong unstable manifold of the non-hyperbolic periodic point with any of the other stable or unstable manifolds, then there is arbitrarily near { ~} an arc { %,} and a ~ ~ R near b such that ~ has a non-transversal intersection of a stable and an unstable manifold of hyperbolic periodic orbits. Again by the results of section I this implies {r and hence, {~} is unstable. Finally, we indicate why q~ r MS for near b and tz # b. Since % has no cycles and the limit set of % has only finitely many orbits, the Birkhoff center of Sb must be finite. Then the arguments in [i5] show that the limit set of q~b equals the Birkhoff center, and, hence, must also be finite. Because Sa is elementary, it then follows that q~, ~ MS for ~ near b and V # b. With arguments similar to those in the preceding paragraph, one can show that a mildly stable arc in d must lie in ~. There are, however, left stable arcs in .~r which are not in ~g: % might, for example, have two Hopf points or flips. All such arcs are left conjugate to left stable arcs in ~. 3" Endomorphlsms of the drcle In this section we first extend the notion of rotation number, defined by H. Poincar6 for diffeomorphisms of the circle, to endomorphisms of degree one. Instead of a number we get in general a closed interval of the real line, which we call rotation set. These rotation sets are then used to analyze a class of I-parameter families of endomorphisms with non-degenerate folds (see (b) below). We show that each such family must go through homoclinic trajectories with folds (see Theorem (3.7)). This result has a direct application to the bifurcation of diffeomorphisms exhibiting a saddle-node with one cycle described in the previous section. If the saddle-node has a one dimensional stable or unstable manifold, they must go through a non-transversal homoclinic orbit. An interesting question is if such families of endomorphisms, for which the rotation sets vary, have bifurcation sets of positive measure. For the diffeomorphism case, see [IO]. A) Rotation sets for endomorphisms of the circle We identify the circle S 1 with R/Z. By End(S) we denote the set of continuous maps 9 : S 1 ~ S 1 of degree i. On End(S 1) we use the usual C o topology. For each 43 S. NEWHOUSE, J. PALIS AND F. TAKENS eEnd(S t ) there is a ~:it-+it such that r~=~ where 7=:it--->S t--->R/z is the canonical projection. ~ is called a lift of 9 and it is unique up to the addition of an integer. Each such ~li satisfies ~(x + i) = ~(x) + I. For 9 t End(S t) with lift ~ and x ~ R we define the rotation number p(~, x) to be lim sup-X (~"(x) --x) and the rotation set p(~) to be closure {p(~, x)[x eit}. n~o n Note that if we take a different lift, say ~, of 9 then p(~, x) and p(~, x), and hence also p(~) and p(~), are equal up to translation by some integer. If x' ~ It with ~(x') = ~(x) then p(~, x') = p(~, x). Hence, up to translation by integers, p(~, x) and p(~) are invariants of ~, ~(x); if no confusion seems possible we may denote them by p(r ~(x)), O(~). We note that p(~, p) and p(~), p ~ S t, are topological invariants: if h : S t ~ S t is a homeomorphism, then O((1),p) ---- p(h*h -t, h(p)) and p((l)) = p(h(I)h-X). Finally, if q) is an orientation preserving homeomorphism, then p((I)) is the usual rotation number [29]. Lemma (3. 9 ). -- If dp t End(S t) and ~) is a lift of r and if there is no periodic point of r with rotation number P-, p e Z, q t N, i.e. if there is no x t It with r = ~(x) and p(~,x) P, then p(~) is contained in { x t it , x < P } or in {xtR[x>P} Proof. ~ If, for some x e It, ~(x) -- x = p then ~(x) is a periodic point with rotation number -P- ; so this does not happen. Hence either ~q(x) -- x < p for all x t It or ~q(x)- x > p for all x e It. Since ~q(x)- x is periodic in x (with period I) there is some ~ > o such that ~(x) -- x < p -- r for all x e R or ~(x) -- x > p -k- r for -- q q Corollary (3.2). -- If 9 tEnd(St), a, ~ tp(~) and ~<P-< ~ for some rational number p, then 9 has a periodic point with rotation number p-q and hence P-q ~ p(~). Since p(~) is closed (by definition), p (~) must be either a single point in It or a closed interval. Now we want to show that p(*) depends continuously on (1). For this we introduce the following notation: if 9 t End(St), ~ a lifting of~ then pl(~), p~(~) t It are such that p(~) = [p~(~), p~(~)]. Proposition (3.3). -- Let UCEnd(S 1) be some open set such that there is a continuous mapping 9 -+ ~ which assigns to each 9 a lifting r (in order that ~ depends continuously 44 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS on ~, U should not be too big). Then the functions gP ~ px(~li) and ~ w, p~(~) on U are cont/nuous Proof. -- We observe that for any rational number P, we have that P- q < pt(~) is equivalent with ~q(x) -- x :> p for all x e [o, x] which is an " open condition ", i.e. the set of 9 E U with pt(~)~P- is open. Analogously, the set of those 9 e U with pa(~) ( P- is open. e (pl(~), pa(~)) if and only if for some big N e N, there are x,y E [o, i] Finally, p with ~sq(x) --x~> N.p + i and ~'q(),) --y( N.p-- i. Also this condition is open, hence pl(~) and ~(~J) depend continuously on ~. Proposition (3-4). ~ Let r e End(S t) with lifting ~ and let P- e (01(~), [~1(~)). Then there is a periodic point s ~ S t of 9 with rotation number P-- such that W~(s) = S x, where Wg(s) --= 9 [U r the intersection being taken over neighborhoods U of s. "EN Proof. ~ We may assume that P- = o (if not we replace 9 by r and choose an appropriate lifting). Let X denote the projection of the fixed point set of ~ in St; since X is closed and non-empty, S ~- X consists of open intervals. Let U be such an interval and U = (~-t, ~) a lifting of U. Then ~(x) -- x, for x ~ U, is always positive or always negative. In the first case, U C W~(n(s-t)), in the second case, U C W~(=(~)). If 9 ~"(U) =S ~ we may take s=r~(s-1) or s=~(s~) and are done. If on the other n 0 hand for each component U of S 1 -- X, ,>~0~"(U) + S t, then r x]) C [-- ~, 2] for all n>o, so we would clearly have p(r ={o}. This would be against the assumptions. Remark (3.5). ~ Let ~ C End(S t) denote the subset of those 9 e End(S 1) which are G t, whose first derivative is of bounded variation and whose critical set consists of a finite and non-zero number of generic folds, i.e. points where 9 is locally C ~ conjugate with the map y = x ~. By a theorem of Block and Franke [2], every 9 in ~ has a periodic point. This is equivalent with saying that if 9 E ~, p(cP) cannot consist of only one irrational real number. This together with the continuity of Px and p~ implies that if R ~ ~ ~-~ r is a continuous arc in End(S t) with image in aj then p(~o) is constant (as function of or) or, for some ~, p(O~) has a non-empty interior. This observation will be useful in the next section. Finally, we pose the following conjecture which extends Herman's result for arcs of diffeomorphisms [i o]. Let { q~} be a C t arc of C ~ endomorphisms of S 1 with rotation sets { [Pt(D), P2(g-)] }- 45 s. NEWHOUSE, J. PALIS AND F. TAKENS Conjecture (3. fi ). -- If the rotation set of r varies with tz, then the set of t~'s for which either Pt(~) or P~(t~) is irrational has positive (Lebesgue) measure. B) Some one-parameter families of endomorphisms of S 1 In this sub-section we consider continuous (or smooth) families ~o, a z IR, of endomorphisms of S t such that for each ~ ~ R, r E ~, see Remark (3.5); the rotation set p(~o) is non-constant, as a function of a; -- for each rational number P, there is a locally finite set Zp C R such that for a r Zp, all periodic points of ~o with rotation number p- are hyperbolic. We shall prove for these families: Theorem (3.7). -- For % as above there are an interval (6t, %) C R, a hyperbolic periodic point s o , a t < a< % of ~o such that W~o(so) is S t for all a ~ (at, %) and a point to, a t < ~ < ~ in the critical image of r (for some n) such that: both s o and t o depend continuously on 6; the curves ~ ~ (so, a) and a ~ (to, a) in S t � (at, a,) cross one another; if ~o is a lifting of ~o, depending continuously on a, and if so is a lifting of so, then t o is the projection of an end point of the interval ~(~o,-so + i ). Remark (3.8). -- For 9 e End(S1), with lifting ~ one can construct a I-parameter family of endomorphisrns q)o by putting q)o(=(x)) = ~(~(x + a)). We note that in the set of (3' endomorphisms ~ of S t, r > 2, which are not diffeomorphisms, there is a residual subset for which the corresponding one parameter families of endomorphisms satisfy the assumptions in Theorem (3.7). Proof. ~ Since p(~o) is non-constant, there is some a3 such that p(~o,) has interior points, see (3.5), and such that p(~o) is not locally constant on a neighborhood of %. Choose a rational number P- in the interior of the rotation set. Since in any neighborhood of % there are infinitely many points where the function p(~o) is not locally constant, we can choose at such that ~o, has a hyperbolic periodic point s with rotation number P- such that W~o,(S) = S t (see (3.4)), and such that p(~o) is not locally constant at at. Now if follows that there is some neighborhood U of a4 in R and a continuous function U ~ a ~ so e S t such that, for a E U, so is a hyperbolic periodic point and such that W~o(So) = S t. Next, since p(~o) is not locally constant at r there is a ~ in U for which p(q~o,) # p(q~). Then one can find two rational numbers Pt P2 with Pt < a0~ such that 46 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS Pl P2] the closed interval ~, ~ is in (O(r -- p(~)) v (p(~) -- p(~o,)). We suppose ~,~ is in 0(~o,) -- P(q~) since the other case is similar. Then, with glio, ~-o liftings of q)o, So, + + = + 3Pl, + 3P ] and _~ ~, ~-~ + x) C complement of [s-~ + I + 3Px, s-~ + 3Pa] ; see also Lemma (3.1). This means that one of the endpoints of the interval O~(s-o, ~-o + i), cr between ~ and ~, has to cross over the interval [~-o + I + 3Pl, To + 3P~]. Since this last interval has length at least 2, one of the endpoints of O~'(~-o, ~-o + i) crosses, after projection on S ~, over So when ~r goes from ~ to 7. Finally we have to show that if, for ~ between cr~ and 7, one endpoint of O~'(~-o, ~-o + x) lies in [~-o + I + 3Pl, To + 3P~] then this endpoint is in the critical image of =o~S~'. To show this, it is enough to show that neither ~'(~-o) nor O~'(~-o + I) can be in ~-o + i + 3Pa, So + 3Ps]- But this follows from the fact that the rotation number of So is P- and that P- does not lie between P x and p~ (because otherwise P- r p(Oo), see (3-~)). q q qx ql q Now it is clear that we can choose (~, ~,) to be an interval between e~ and ~ such that if ~ goes from ~x to ~, one endpoint of ~'(~-o, S-o+X) crosses over the interval [~-o + x + 3Pt, To + 3Pa]. We choose then n = 3ql and we choose to to be the pro- jection of that endpoint. C) Application to bifurcation theory The analysis of families of endomorphisms in the last section leads to results also when analyzing certain families of diffeomorphisms. An explicit formulation of what we need in this direction is: Remark (3-9). -- Let ~o : S 1 ~ S l be a one-parameter family (r e R) of endo- morphisms as in Theorem (3-7)- Let ~o,~ : S 1 � [o, i] ~ S 1 � [o, i] be a 2-parameter family of C ~ maps depending continuously on (a, ~) such that: -- q)o,0(x, t) ---- (q~o(X), To(x)) for some function To; -- for ~z positive, 0o. ~ is a diffeomorphism into. Then, for ~ sufficiently small, there is a a~ E (el, ~,) such that *~g.~ has a non. transversal homoclinic point. Even more holds: for ~ sufficiently small, and any curve (a,f(,)) with o <f(a) < ~, e (*l, as), there is a ~/such that r ) has a non-transversal homoclinic point. The proof of this statement follows from the following continuity considerations. The map q)o.~ has a hyperbolic periodic point so,~ = (Xo.~, to,~) such that Xo, 0 = so 47 S. NEWHOUSE, J. PALIS AND F. TAKENS as in Theorem (3.7). We observe that W'(so,~,) depends continuously on (~, [z) and W*(s,.0) = {Xo,0} � [o, I] (note that % : S 1 -~S 1 was expanding at So). One can choose a local unstable manifold W~o0(So.~) as an embedding (or its image) which depends continuously on (~, ~). For some k, k ?o,0(Wl0~(Xo,0) ) has a point To in its critical image which moves over So, 0 by Theorem (3.7). From this and the above continuity we see that, for each small enough tz > o, there is a value of ~ such that k is not transversal with respect to W"(so,~). 4. The saddle-node with x-cycle in dimension 2 Let {?~}: M -+ M be an arc of diffeomorphisms of a compact 2-manifold M so that for ~ < b, % is Morse-Smale and such that for [z = b, % has a saddle-node at p (we assume p to be a fixed point of % ; the case where p is periodic can be handled analo- gously). Further we assume that % has a x-cycle containing p; i.e. there are non-trivial intersections of W"(p) with W'(p) and there is no periodic point q such that both W~(p) ~ W'(q) and W~(q) n W'(p) are non-empty. Theorem (4. 9 -- Under the above hypotheses { % } is not mildly stable. Remark (4.2). -- It will be evident from the proof of Theorem (4. i) that the same result holds if dim 1V[ > 2 and % has a saddle-node in a l-cycle whose stable or unstable manifold is one dimensional; i.e. a normally repelling or attracting saddle-node. An open dense set of these arcs will create homoclinic tangencies. Thus, in dimension two they will contain diffeomorphisms with infinitely many sinks or sources [2~]. Proof of Theorem (4. l ). -- An arc { %} is already not mildly-stable if in any neighbor- hood of % there is a non mildly-stable arc. Hence we may, without loss of generality, impose generic conditions on { % }. In particular, we assume that p unfolds generically. Also, we may assume that the eigenvalues of (d%)v are i and ~r with o < ~ < i ; in case 0t> I we take?~-l, incase -- i< 0t<o we take ?~ and in case 0t<-- I we takerS2 instead of 9. From the fact that, for ~ < b, ?~, is Morse-Smale and the fact that {%,} unfolds generically at p, we may assume that, near p, ?~ has two fixed points for tz < b, one fixed point for ~z ----- b and no fixed points for ~z > b; we also assume that W~(P, ~b) n 0(W'(p, ~b)) = {p}. We first observe that W"(p, %) C W'(p, %); this follows from the following two facts: -- W"(p, %) -- {p} t~ W'(p, %) is open in W~(p, %) -- {p} and non-empty; -- for any filtration {M~} of M for ~b (see [20]), with p e Mi. -- M~._I, the set of points in W"(p, ?b)- {P}, which go eventually into Mi_ 1 is open and equals (w-(p, -- {P }) -- W'(p, 48 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS In the case that W~(p, %) closes to a smooth circle, p has a non-critical I-cycle. Then the instability here was proved in Proposition (2.3). Thus, we may suppose W~(p, %) does not close to a smooth circle. This means that the projection of W~'(p, %) on a center manifold in W'(p, %) along the strong stable foliation does not have maximal rank everywhere. We shall approximate { %, } by a family {~,} such that non-transversal intersections of stable and unstable manifolds occur for certain ~'s arbitrarily near the first bifurcation of{~'~}. From this, we infer that {~"~}, and, hence also {%,}, is not mildly stable. This will prove Theorem (4. I). For convenience of notation, let us assume that { %, } is defined for Ez near o and that Ez = o is the bifurcation point instead of ~ = b. For each diffeo- morphism ~'~ we require that there is a smooth vector field X~, defined for ~z close to o on a neighborhood of p, such that the time x map X~. x of X~, wherever defined, equals ~. Also we require that there are smooth coordinates y, z (which may depend on ~) such that X~, locally has the form 0 0 X~ = Y~(y) ~y + Z,(y).z. Oz O~Y~176 OY ~ > o, and { Y~,(y) ~y } where Y and Z are smooth functions of (y, Ez), Zo(o) < o, is a saddle-node arc. The fact that such ~"~ exists arbitrarily close to (p~ follows from [38], [39]. Choose, for ~ > o a fundamental domain D2.~ for ~ in the positive y-axis, smoothly depending on F~. For some big m, ~'~(D2,0) will again be in our coordinate ~o ~ O~(D~. o) ,~, ', D ,o D2,0 z< Flo. 8 S. NEWHOUSE, J. PALIS AND F. TAKENS 5o neighborhood the same holds for ~"(D2,N) if ~x is near o. Define D1, N to be the funda- mental domain in the negativey-axis whose boundary is the projection, along the z-direction, of the boundary ~" of %, (D2,N). We take a " rectangle " ~N SO that its boundary consists of two pieces of XN-integral curves and two pieces of straight lines parallel to the z-axis, such that its projection on they-axis is D1,N, and such that for each q e ~'~(D2,N) , there is a positive m' such that ~'(q) is in the forward orbit of 9,,. See figure 8. Let A N : 9N "--> M, ~ > o be defined as follows: Ao(~o) = D2,o and Ao(y , z) = (ao(y), o) where a 0:Dx, o~D2'0 satisfies (ao).X o=X o on {z=o}. For ~>o there is a positive T N such that the time T N map of XN, (XN)~N, satisfies (XN)~ND1, ~ = D~,~; we define A N to be (XN)r. For e e [o, I] we define Ao,~ ----= (XN) o oA~,. Next we define Bo, N ----~' o Ao,~. For ~ close enough to o, the image of Bo,~ will be in our coordinate neighborhood in the part {y < o }, and also there is some posi- ~.,, tive m' such that the image of ~ o Bo,~ is in the forward orbit of ~N" Consider now the quotient map {(y, z) [y< o} -+ {(y, z)[y< o}/(y, z) '~ ~N(Y, z). Under this quotient, ~, becomes an annulus ~, Bo, N becomes a z-parameter family of mappings Bo, N of ~N into itself and the projection ~(y, z) = (y, o) goes over into ~, the annulus projection. We want to show that Remark (3.9) is applicable to BO, N" In order to prove this we need new coordinate functions s~:~ N -+ R/Z and w~, : ~N ~ IR such that ~(sN, w~,) = (s~,, o) and such that X~, has the form --. We Os N shall write s, w instead of s0, w 0. For some circle endomorphism ~ : R/Z -+ R/Z and some map W : R/Z ~R we have B~ w) = (~(s -k ~), W(s)). The circle endo- morphism ~ is determined by ~'0; ~'0 was obtained from % by a small, but otherwise arbitrary, perturbation. Hence we may, and do, assume that the family { ~o } defined by ~o(s) = ~(s + ~) satisfies the assumptions of Theorem (3.7); see also Remark (3.8). Now it is clear that Bo, N satisfies the assumptions in Remark (3.9). Now we observe that if T~, q- ~ eN, BO.N is an iterate of~, (up to the identifi- cations). Hence if such a Bo,~ has a non-generic tangency of a stable and an unstable manifold then the same holds for ~'N" Also, Bo,~ satisfies the conditions of ~o,~, in Remark (3-9). Let (~i, ~2) and ~ be as in that remark. Now as ~t -+o, we have T~, -+ oo. For each a ~ (~ri, ~) choose anf(~) such that TI(o) + cre N, and f(a) < "~. We may arrange for a ~f(~) to be continuous on (cl, ~2). Then as in Remark (3.9), there is a at ~ (at, ~,) so that Bot.t(otl , and hence ~~ has a non-transversal homo- clinic point. 50 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 5" On the rigidity of the nnfolrl;ng of the saddle-node In Chapter II we have seen that a conjugacy between two Coo saddle-node bifur- cations must satisfy very restrictive conditions. In particular, at the bifurcation parameter it must be C Oo along central curves away from the fixed point, and it must preserve adapted saddle-node vector fields along these curves. In the present section, we shall present two more applications of the restrictive nature of such conjugacies. The first concerns certain arcs of diffeomorphisms of the circle, and the second concerns arcs between Anosov diffeomorphisms and so-called DA diffeomorphisms. These furnish more examples where mild conjugacies cannot be strengthened to conjugacies. A) One-parameter families of diffeomorphisms of S 1 We consider one-parameter families q~:S1---> S 1 of Coo-diffeomorphisms of S 1 with rotation number p(~t) increasing such that whenever 0(5) is rational, q~g has two hyperbolic periodic points or one periodic point of saddle-node type which unfolds generically and such that whenever p(~) is irrational, p is not locally constant in ~. Let ~'~, : S I --> S 1 be another such family of diffeomorphisms with rotation number p'(~). If we assume that Image(p) = Image(9'), then there is a homeomorphism h : R -+ It such that p'(h(~)) = p(~t), see [5], [IO]. For each ~, r and q~,) are now conjugate: for p(~t) irrational this is Denjoy's theorem [6], for p(~t) rational it follows from the above description. So it is clear that the above two arcs are " mildly conjugate " A question, which was raised independently by R. Thorn and S. Smale, is: are two arcs as above in general conjugate? We show here that the answer is in general negative. Let {%,}, {~0~} be a pair of arcs of diffeomorphisms of S 1 as above. Let ~. be a boundary point of p- 1 (~) for some rational P- q e Image(p). If there is a conjugacy (h, H~) between % and %, i.e. h : It ~ It a homeomorphism and H~, a conjugacy between % /L't i ouo y o ntof,, .ow and ~)depending continuously on ~t, then h(~) q~ has a unique fixed point which is of saddle-node type. By Chapter II, Theorem (3. ~), there is a unique smooth vector field X near the saddle-node orbit such that its time one map X x equals (~g)q (and such that (~g).~X = X). Also for q~h(~,l there is such a vector field X'. Again by Chapter II, Theorem (3.2), Hg has to map X to X'. If we now extend X, and X', to all of S t so that ~g.X = X, q~(~l.X' = X', we obtain in general bivalued vector fields. This means that, on the complement of the periodic orbit, Hg has to respect two different vector fields. This is in general impossible. B) Arcs between Anosov and DA diffeomorphisms In [34], S. Smale showed that certain Anosov diffeomorphisms may be modified to give Axiom A diffeomorphisms with attractors having intricate topological properties. 51 S. NEWHOUSE, J. PALIS AND F. TAKENS Since the latter diffeomorphisms were obtained by modifying Anosov diffeomorphisms, he called them DA (for derived from Anosov) diffeomorphisms. Although Smale's construction was given for Anosov diffeomorphisms of the two torus, it works just as well with codimension one Anosov diffeomorphisms of the n-torus. A brief description of the construction is as follows. Let L be an Anosov diffeomorphism of T n and suppose that dim W=(x) = I for each x e T ~. Since L is conjugate to a linear toral automorphism [I6], it has a fixed point, say p. Assume that the contracting eigenvalue ofp is positive. Locally, near p, one has the usual picture of a hyperbolic saddle fixed point as in Figure 9 a. P P~ P2 \/ \ / FIo. 9 a FIO. 9 b FIO. 9 Smale proposed to modify the diffeomorphism L in a disk neighborhood N of p to obtain a diffeomorphism g with two new saddle fixed points Pl and p, on W=(p, L) and such that p is a fixed source ofg (see Fig. 93). This can be done so that g agrees with L off N, g satisfies Axiom A, and the non-wandering set of g consists of p and an (n -- I)- dimensional hyperbolic attractor containing PI and P2. In fact, as Williams pointed out in [42], one can choose g so that the foliation #'= = { W=(x, L) I x ~ T n) is g-invariant. Of course, the unstable L-foliation is no longer g-invariant. Somewhat later in a private communication with us, Williams observed that a I)A diffeomorphism g could be cons- tructed from an arc in which a saddle-node occurs. The local picture is in Figure 9 c. // // .... // t// Fio. 9 r If one chooses such an arc carefully, then one can actually make the arc mildly stable with an isolated bifurcation point. This will be proved elsewhere. These mildly 52 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS stable arcs are such that at the bifurcation point one has a saddle-node p whose stable manifold W'(p) is one dimensional and such that there is a hyperbolic periodic point q such that W~(q) c~ W~ + o. The next proposition shows that such an arc is never stable. Proposition. -- Suppose { %,}, o < ~t < I, is an arc of diffeomorphisms of T" so that ~o is Anosov with dim W~ = I for all x. Let b = b({%,}) be thefirst bifurcation point 0f{9~} and assume o < b < i. Suppose that % has a saddle-node periodic point with dim W'(p) = x and ~b has a hyperbolic periodic point q not in the orbit of p such that W"(q) n W*(p) + o. Then {~} is not stable. Proof. ~ We may assume, by perturbing {%,} if necessary, that W"(q) is transverse to W'(p). Since q can be continued to a hyperbolic periodic point q~, for ~t<b, and %, is Anosov for ~t<b, it follows that dimW"(q~) =dimM-- t. As W*(q) n W~ 4= o, this intersection is zero dimensional, and, therefore p is s-critical. Now the method of proof of Proposition (2.4) may be applied to show that {%,} is not stable. C) One-parameter families of vector fields As was pointed out in [4I], the rigidity in the conjugacy of a saddle-node arc has consequences for the stability of one-parameter families of vector fields. Consider such a C | one-parameter family{ X~,}on a 2-manifold M such that for ~ = b, X b has a saddle- node closed orbit y, i.e. a closed orbit which is attracting at one side and repelling at the other side and whose PoincarE map has first but not second order contact with the identity. We assume furthermore that "f unfolds generically; by this we mean that if S is a local cross section ofX v intersecting y, then the Poincar~ map P~ : S -~ S, ~t near b, is a saddle- node arc, see section 3 of Chapter III. Let {X~,} be another one parameter family of vector fields on M (near {X~,}) so that for ~t = b', X;, has a saddle-node closed orbit y' which unfolds generically. Let S' to be a local section of X;, intersecting y'. We assume for simplicity that neither {X~,} nor {X~} has any other saddle-node closed orbit. We say that these two families are topologically equivalent if there exist a homeomorphism h : R --~ R and a homeomorphism H~, : M -+ M, depending continuously on ~t, such that H~ maps integral curves of X~ to integral curves of X;~,). If such an equivalence (h, tt~,) exists, then h(b) = b'. Modifying H~, along the orbits of X, we may assume that H~(S) = S' for bt near b. So H~ ] S has to conjugate the saddle-node arc {P~} with {P~J. By Chapter III, section 3 this implies that Hb] S has to map X to X', whele X, X' is the unique smooth vector field on S, S' such that its time I map X1, X' l equals Pb, P;'. This means that Hbl S is determined as soon as it is determined in two orbits of Pb in S, one on each side of y c3 S. Since H b has to map separatrices, i.e stable or unstable manifolds of saddle points, to separatrices, the map Hb[ S is essentially fixed for each intersection of S with a sepa- 63 54 S. NEWHOUSE, J. PALIS AND F. TAKENS ratrix; such an intersection is an orbit of Pt. So in general {X~,} and {X~} will not be topologically equivalent if at least two separatrices approach ~" from the same side. This means that an arc { X~,} which has a saddle-node closed orbit which is approached from the same side by at least two separatrices is not stable; it has a modulus of stability in the same way as arcs of diffeomorphisms with a tangency of stable and unstable manifolds. In [7] Guckenheimer incorrectly states that some of these arcs are stable. 54 IV -- GLOBAL STABILITY 9 . Introduction In this chapter we complete the proofs of our main results concerning stability of one-parameter families (arcs) of diffeomorphisms, stated in the introduction to the paper. Necessary conditions for stability, mild and left stability at the first bifurcation point were provided in the previous chapter. We now show that these conditions are also sufficient. As a consequence we obtain a characterization for the stability of arcs containing several bifurcation values, under the basic assumption that the limit sets have finitely many orbits. The main idea here consists of a suitable construction of tubular families or folia- tions, which will be used to define topological conjugacies. We begin by describing local tubular families for hyperbolic periodic orbits. This concept will then be extended to Hopf orbits, saddle-nodes and flips. The foliations are constructed for a family { %} in ~, as defined in the introduction, especially near its first bifurcation value b. In the case of left and mild stability, this interval is of the form [~1, b], for some ~x ~ b. Other- wise, it is of the form [tzl, g~] with b E (Ezl, g~). For each Ez in such intervals, we build up tubular families or foliations requiring them to be %-invariant. We shall usually assume that the periodic orbits of % are fixed points. In fact, if a periodic orbit of % has period k, we can consider ?~ to define the foliation near one of its dements, and use %-iterates of the leaves to obtain the foliation near the others. 2. Local Tubular F~m;l;es We first recall, in a parametrized version, the notion of tubular family for a hyper- boric orbit [24], [~5]. Let ~- ~ M be a hyperbolic fixed point for r ~ ~ I. Let [~1, g~] be a neighbor- hood of ~ in I and U be a cell neighborhood of s in 1VI. If these neighborhoods are small enough, there exists a continuous mapping [[zx, ga] ~ ~ ~ x~ e U, where x~ is the unique (hyperbolic) fixed point for % in U and s x~. We denote by W"(x~,) and W'(x~,) the unstable and stable manifolds of % at x~. Given the family ( % }, V e I, we define (I) : M x I --* M by (I)(x, ~) = (%(x), tz). Definition (~,. 9 -- An unstable tubular family for { % } or 9 at (~, ~) is a continuous foliation ~" of U � [[zx, g,], such that for [z ~ [~1, ~] a) the leaves are C ~ discs raring continuously in the C k topology, I <_k< 0% and = x { } n u x { }, 65 S. NEWHOUSE, J. PALIS AND F. TAKENS 5 6 b) each leaf o~'~(y, ~) is contained in U � { ~. } for y ~ U, c) the foliation is O-invariant in the following sense O(o~"(y, p.)) D .~"(%,(y), p.) for y, ~(y) ~ U. Remark. -- For our purposes it is enough to take the leaves of the foliation to be C 2. In fact, in their global setting, we can and will construct the foliation to be C * (k > 2) when restricted to any unstable manifold of a hyperbolic periodic orbit that intersects W'(x~) x { Ix }. See Definition (3. x) of the next section, where even more differentiability is involved. The same remark applies to Definition (2.2) below. Let us see how a similar foliation can be defined at a Hopf fixed point p of %. We assume, for tx < b and tx near b, that q~, has a hyperbolic fixed point p~ near p and dim W'(p~,) = dim W'(p). As ix -+ b, ix < b, the stable and unstable manifolds of p~, in U converge to those ofp in the C k topology. So, we can use Definition (2. i) in this case, with the difference that we consider tx in some interval [~x, b]. As a natural extension of Definition (2. i ), Let nowp be a saddle-node or aflip for %. we present below the concept of strong unstable foliation for { %, } or 9 at (p, b). First we need some basic facts about center manifolds as stated in Chapter II. Let U be a small cell neighborhood ofp in M and [~q, ~ C I a small interval with b e (~q, ~). Let W ~ bea C ~center manifold for 9 at (p,b), I<k< oo. For each ~tr tJv.], W~, = W c n U x { ~ } has dimension one and W~ is C ~176 We also consider the center stable manifold W cs for 9 at (p, b), which is C k and has dimension s + 2. Here s and u are the number of eigenvalues (with multiplicity) of d%(p) with norm less and bigger than one, respectively. Both W ~ and W ~ are invariant by 0, and W~, and W~,' = W ~' t3 U � { ~t } are invariant by r Frequently, here and in the sequel, we will identify a subset V � of M � with its projection V into M. We recall that W ~ is foliated by the strong stable foliation ~'"s, with leaves s-dimensional C k discs transverse to W ~. Particular leaves are the strong stable manifolds through the fixed or periodic orbits of % near p. For each ~ e [t~l, ~], W~,' is a union of leaves of ~~'. The foliation ~-" is invariant by 9 in the following sense: if z, 0(z) z U X [~1, t~] and S is the leaf through z, then 0(S) is contained in the leaf through 0(z). Similarly, we can define the center unstable manifold W ~" for 9 at (p, b) and W ~" is foliated by the strong unstable foliation #-". Particular leaves are the strong unstable manifolds of the fixed or periodic orbits of %, near p. Our strong unstable tubular family (or foliation) is an extension of ~"" to a full neighborhood of (p, b) in M x I. For later purposes we need this extended version of ~'", but no such version of #'". Suppose (p, b) is a saddle-node. As it can be seen from Chapter II, there are two possibilities for its unfolding. In the first one, for each ~ < b and near b, there are two hyperbolic fixed points Pl,~, and P2.~, of ~, near p, which collapse into p as tt -+ b and then disappear for ~> b. We also may assume that dimW'(p,.~) = s + x and g6 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS dim W~ = s. The other possibility is similar, only the orbits Px,~ and P2,~, appear for ~ > b. Throughout this chapter we assume the first case. If (p, b) is a flip, we also assume the following of four similar possibilities for its unfolding. For each IX < b and near b, there is a hyperbolic fixed point p~, near p. For ~x ~> b, there is a hyperbolic fixed point Pl,~, and a hyperbolic period two point P2,~, near p. We assume that dim W'(pl,~) = s and dim WS(p2,~,) -= s + I. Definition (~,. ~). -- A strong unstable tubular family o~ ~' for { % } at the saddle-node or flip (p, b) is a continuous foliation of U � [~x, ~] such that a) the leaves are C k discs varying continuously in the C k topology and b) = W"(p, b) n U x { b }, b) for each ~, U � { Ix } is a union of leaves transverse to W~,', c) the foliation is ~-invariant: if o~(y, ~) is the leaf through (y, ~) eU x {~}, then r ~)) is the leaf through (?~(y), ~) ~U � These local unstable and strong unstable foliations have already been used in a similar context by several authors; see [26] for references. We provide a construction of them in Proposition (2.3) below, to give a clearer view of some of the main techniques of this chapter. Proposition (2.3). -- There exists a strong unstable foliation for {%} at (p, b), where p is a saddle-node or a flip. Similarly, there exists an unstable foliation at a hyperbolic or a Hopf fixed point. Proof. -- Let us first consider (p, b) to be a saddle-node or a flip. In U � [[xa, ~ua] let W ~' be the center-stable manifold of q) at (p, b). As before, 9 is defined by 9 (x, Ix) = (%(x), ~) and U ~p, [~xl, ~] ~b are small neighborhoods. In W ~ we take a closed fundamental domain A. This is a set with the following property. ~](A) nA=o and there is a neighborhood U' of (p,b) in W ~* such that if xeW ~~ and its negative q)-orbit leaves U', then the ~-orbit of x has an element in A, which is unique except if it belongs to 0A. Let us indicate the construction of such a fundamental domain when (p, b) is a saddle-node. Taking a C z coordinate system in the center- stable manifold of ~b at p, we can write %(x,y) = (x + x 2 -t- x(c,y) -t- r 3, [I xay 1[, [ly [t2),Ay + ~(11 xy I[, [[y][2)), where c e R 2 and []A[] < i. Here y -= o represents the one-dimensional center- x = o the s-dimensional strong stable manifold W*'(p). The stable manifold and manifold W'(p) of % at p is the half-plane x ~ o. We take in W"(p) a hemisphere H of small radius centered at p and slightly extended transversaUy across W"(p). It is easy to check from the above expression that %(H) nil---- e. For Ix1 <_ ~_< b, H~ = H X{~} also crosses W"(px, [z) and W'"(p2, Ix) transversally in U � {ix} and 8 S. NEWHOUSE, J. PALIS AND F. TAKENS ?~(H.)c~H~=o. Also, for b<lz<~, ?~(H.) c~H~-----e, where H~=Hx{Iz}. We let A~ be a closed region bounded by H., ?~(H.), and an annulus whose boundary lies in H~ c~ ?~(H~). Then take A = U A~. ~,<~<~., Ap Ao ~l (HjL) ~b (H) ~~X~-J Hy ~H Fio. Io In the case of a flip, A is diffeomorphic to A b � [[zl, ~,], where A b is an s + l-dimensional annulus. Similarly, if (p, b) is a Hopf point, A is diffeomorphic to A b � [~Zx, b], where A b is an (s -t- 2)-dlmensional annulus. If (~-, ~) is a hyperbolic fixed point, A is diffeomorphic to A~ � [~zl, g~], where A; is an s-dimensional annulus and ~ ~ (~1, ~). Let us proceed with the construction of the strong unstable foliation for a saddle-node or a flip. Over a neighborhood U1 of the exterior boundary of A we raise a fibration of class G k, the fibers being u-dimensional discs transverse to W c' and each of them contained in some U � { ~ }. Over the neighborhood O(U1) of the interior boundary of A, we have a similar fibration, the fibers being the O-images of those in U1. Restricting to smaller neighborhoods of ~A, we can extend this fibration to a full neighbor- hood of A. This is done as follows. Over a neighborhood of A, we raise another C k fibra- tion transverse to W ~' and the fibers contained in the sections U � { ~ }, but not necessarily %,-invariant. We now define a new fibration, which agrees with the first one in a neighborhood of ~A and with the second one off a slightly bigger neighborhood of ~A. Let ~1 and ~2 be the projections, into a neighborhood V of A, defined by the two fibrations. We define a C k real function p : V ~ [o, I], such that p is I near ~A and o off a small neighborhood of ~A. The required fibration is then given by the projection = Pnx + (I- P)n2. Its fibers form the leaves of our foliation. We now simply define it over W ~ -- W ~ through iterates O" or V~, all n~ o: if (y, F) -- O"(x, tz) for (x, ~) E A and the fiber through (x, ~z) is ~""(x, ~z), then the fiber through (y, ~) is O"(.~~"(x, F)) c~ U � [[zx, ~]. By the generalized k-lemma [24], [3o], ~-" extends over W ~ satisfying all the conditions of Definition (2.2). Notice that, on the center unstable manifold of 9 at (p, b), we get the usual strong unstable foliation. The cons- 58 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS truction at a hyperbolic fixed point is easier and can be done in a similar way. The same applies to a Hopf point (p, b), in which case we take the parameter ~ in some interval [~l, b]. This finishes the proof of the proposition. We point out that these unstable and strong unstable foliations are not unique. There is a degree of freedom in their construction, as shown in Proposition (2.3). This is what enables us to globalize them in a compatible way with the tubular families of other periodic orbits. Such globalizations will be performed in the next section. 3- Global Tubular FamIHes In this section we construct compatible systems of tubular families or foliations for a family {%,}, ~z near its first bifurcation point. As before, we denote by b e I the first bifurcation point, so that % is Morse-Smale for tz ~ b. From the previous chapter, {q~,} can be stable in one of our three senses only if one of the periodic orbits of ~b is an elementary bifurcation (saddle-node, flip or Hopf). Moreover, the (strong) stable and unstable manifolds of all the periodic orbits of % must have transversal intersections. We recall that, since q~ is Morse-Smale for [z ~ b, its periodic orbits can be partially ordered through the relation pi,~>pj,~, if W~(p~,~,) nW*(pj.~)oe o. See [24], [25]. We fix a total ordering for the periodic orbits compatible with this relation. If ~b has a Hopf orbit or a flip, the same ordering applies as well to its periodic orbits: there can not be any cycle, since otherwise b would not be the first bifurcation point [x8]. If 9b has a saddle-node, we may have a cycle containing this orbit. If there is only a I-cycle, the saddle-node will be counted as pj,~ and Pj+t,~, for some positive integer j. However, if there is a cycle of larger length, then the saddle-node will be counted as p~,~, Pj +j,, ~ where j and j' are positive integers and j' > I. In all cases, this ordering of the periodic orbits will be used to build up global systems of foliations for { % }, ~ in a small interval in I. This interval is of the form [~t, b] when % has a Hopf point or a saddle-node which is critical or has a cycle, and otherwise [Eli, ~] with b e (~1, ~). Let Px,~ > P,,~, > 9 9 9 ?~ Pt,~, be the periodic orbits of % for some ~ ~ [~q, g~], all of them hyperbolic except at most one which is an elementary bifurcation. For convenience, we write p~ instead of pi,~. Let Ui, U2, ..., Ut be neighborhoods of Pi,P2,...,Pt. We will consider local unstable or strong unstable foliations in U1 � [~li, ~], U2 � [~i, ~u/], ..., U t � [[ii, g~] as in Definitions (2. I) and (2.2). If the periodic point p~ is a saddle-node with no cycles or a flip, let .,~-~(pi) denote its strong unstable foliation in U~ � [Eza, ~]. Otherwise, let o~-~(p~) denote the unstable foliation in U~ � [#~, g~]. Since ~-~(p~) is a ~;invariant foliation in U~ � [~, g~], it naturally induces one in [.] ~,(U~) � {#} for [z e [#x, g~] and n e N by simply taking iterates ~, of the leaves. In the sequel, we will be using this extended (or globalized) foliation, which will be still denoted by ~z-~(pi). 60 S. NEWHOUSE, J. PALIS AND F. TAKENS Definition (3. ~ ). -- The system of foliations o~'.(pi), o~',(p,), ..., o~',(pt) is compatible when a) ira leafF of ~"(p~) intersects a leafs of ~(p~), i ~j, then FDS, b) for all i ~j, the restriction of ~"(p~) to each leaf of ~(p~) is a foliation of class C ~, ~ ~ k. Remark. -- Note that the restriction of these foliations to unstable manifolds of hyperbolic periodic orbits are of class C *. However, the unstable manifold of a non- critical saddle-node with no cycles or a flip is not a leaf of the strong unstable foliation. Thus, we do not demand differentiability of the foliations when restricted to these unstable manifolds. Let us show the existence of such systems. First we consider the case where ~ has a Hopfperiodic orbit. This is the same as the parametrized version of the hyperbolic ease. Although somewhat simpler than the saddle-node and the flip, it gives a pretty good idea of how to proceed in those cases as well. Proposition (3.2). -- If 9~ has a Hopf periodic orbit, then there exists a compatible system of unstable foliations for ~,, ~ ~ [~l, b] for some ~i near b. Proof. -- Let Pl,~ )" P,,~ > ..- > Pt,~ be an ordering of the periodic orbits of ~ for ~1 ~ ~ ~ b. We simply write p~ instead of P~.b. By induction, we may assume that a compatible system of foliations ~-"(Px), ..., ~-~(P.) has been constructed in neighborhoods U l� ...,U.� [~1, b], p~Ui for I <i<n. Let us build ~"(P,+ 1)- We will adapt the proof of Proposition (2.3) to guarantee the compatibility condition. Near P.+,, we consider a closed annulus A b as a fundamental domain for W'(p,+,) and denote by Oe.A b its exterior boundary. Then, the annulus A~, with boundaries 0exAb � {~} and ~,(O,.A b � {~}) is also a fundamental domain for W~(p.+a, ~), p~ e [~l, b] (we take ~1 closer to b if necessary. If W~(p,) intersects W'(p,+l) it must do so transversally. Thus, we can take a C u fibration of W"(p,) near the exterior boundary of Ab, whose fibers are discs transverse to W~ with the same dimension as W~(p,+,). The image by ~b of this fibration induces a similar one near the interior boundary of A b. On the other hand, it is easy to get a second C ~ fibration of W"(p,) in a neighborhood of all of its intersection with Ab, if we do not require it to be %-invariant. However, as in Proposition (2.3), this second fibration can be deformed to agree with the first one near the boundaries of A n and, thus, it becomes ~b-invariant. We now want to fiber in a similar way the leaves of o~"(p.) near A~,, ~1 ~ ~ ~ b with ~x close to b. First, we observe that, in U, � [~l, b], the leaves of ,~"(p,) are C k imbed- dings of the disc W~(p.) n U,, continuously parametrized by (W'(p,) n U,) � [~1, b]. Thus, by restricting this parameter space, we have that the leaves of ~'"(p,) near A~ are (I k close to W"(p,). Using this parametrization and the fact that the foliation o~'~(p,) is ~-invariant, we can fiber as above the leaves near W*(p,) n A b. We get a ~-invariant fibration with C k fibers transverse to We(p,+l, ~), ~i ~ ~ ~ b, and varying conti- nuously in the fil k topology. By construction, each fiber is contained in some leaf of 60 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6i o*'"(p,). Also, the fibration is C k when restricted to each leaf of o*'"(p,). Now we take #-"(~,-1). There are two cases to consider. If W"(p,_l) intersects W~ but not W'(p,), then W"(p,_ 1) n A b is compact and disjoint from W"(p,) n A b. So, we can proceed as before, fibering the leaves of ~-"(P,-a) near W~(p,_l) n A. Let us now suppose that W"(p,,_ 1) intersects W*(p,). Since o*-"(p,_ 1) and o*'*(p,) are compatible by the induction hypothesis, the fibration of o*"(p,) in a neighborhood V of W"(p,) n A b is also a fibration of o~-"(p,_ 1) in V. On the other hand, B = W"(p,_ 1) n (A b -- V) is compact. So, o*"(p,_a) can be fibered as above in a neighborhood of B. As in Proposition (2.3), we can average these two fibrations near 0B to get a desired one in some neighborhood of W"(p,)u W"(p,_l) intersected with A b. Its fibers form the leaves of our foliation. We repeat the argument to all o*'~(p~), t < i < n -- 2. Once we have the q~-invariant and compatible foliation o*-"(p,+a) near A b � [~1, b], some txx close to b, we just consider its positive iterates by % (or r By the generalized X-lemma [24], [3o], it extends to the foliation W"(p,+ 1.~,), ix 9 [ix1, b], of the center unstable manifold of q> at (P,+I, b). This finishes the construction of the folia- tion #'~(p,+ 1). The proof of the theorem is complete. Let us now consider the case where % has a saddle-node. Let Pt,~, > P2,~ > ... > Pt,~ be an ordering for the periodic points of q~, bit <__ ~ < b. We first consider the case where the saddle node is non-critical and % has no cycles. We assume that pj,~, and Pj+a,~ coalesce at ~ = b, giving rise to the saddle-node pj = Pj+t. We shall construct a compatible system of foliations for {%,}, with ~ in some interval [~t, V,.] and b 9 (~t, t~). The foliation at the saddle-node pj =Pj+t will be a strong unstable foliation ~(pj). Actually, the construction we just performed in Proposition (3.2) can be adapted to the present case as well as to the flip bifurcation. However, to prove the stability of the family, we also construct a one-dimensional center foliation in a neighborhood of the saddle-node pj = pj+ t in the center stable manifold of O at (p~,b). As usual, O:MxI--~M is defined by r ix) = (q~(x),lx). This center foliation o~" should be q~-invariant and compatible with the unstable folia- tions o~(p~), i<j, in the sense of Definition (3. I). Proposition (3- 3). -- If % has a non critical saddle-node p~ = pj + 1 and no cycles, then (a) there is a compatible center foliation ~'~ defined in a neighborhood of the center stable manifold W ~" of * at (Pj.b), (b) there is a compatible system of unstable foliations ~"(pi) for 9 or ~, ix 9 It� ~] for some ix1 < b and ~ > b. The foliation we consider at pj = pj + t is a strong unstable one, o~"", (c) the union of leaves of ~"" through a leaf of o*" forms a C t submanifold. Proof. -- Since Pl > P2 > 9 -- > Pj- 1 are all hyperbolic, Proposition (3.2) provides a compatible system of unstable foliations o*'~(px), ~-"(p~), ..., o~"(pj_l). The leaves of these foliations are taken to be C1 k, k > 2. Let us construct a center foliation ~'* 61 62 S. NEWHOUSE, J. PALIS AND F. TAKENS at the saddle-node; its leaves will be C ~'-~. In the center stable manifold of r at (pp b), we consider a fundamental domain as in Proposition (2.3). We define the center foliation ~'~ as the integral curves of a vector field X satisfying the following properties. X is tangent to the leaves of ~'*(pi) and it is C k- 1 along these leaves for all i < j such that W*(pi) intersects W'(pj). The vector field X is also transverse to the strong stable foliation of~ at p~ and d~-invariant. To get such a vector field, we proceed by induction taking i =j -- I, j -- 2, ..., I. If W~(p~_l) intersects W e', we take a C k-a vector field X tangent to W"(pj_l) near its intersection with the exterior boundary of the fundamental domain A. Near the interior boundary, we just consider d~(X). It is easy to extend this vector field X to a full neighborhood of W~(p~_l) t3 A in W~(p~_l). We now want to define X along the nearby leaves of ~-"(P~-I). As we noticed in the proof of Proposition (3. ~), these leaves are C k imbeddings of a disc in W"(pi_l), conti- nuously parametrized by their intersection with W'(pj_x). So we can project X into the leaves of ~"(Pj-1) near the exterior boundary of A, consider its image by d~ and extend it across A as above. Due to the fact that the saddle-node is noncritical, X is transverse to the strong stable foliation of 9 in the center stable manifold. By induction, let us suppose X defined along the leaves of~'~(pi + 1), 9 .., ~-~(Pj- 1) near the fundamental domain A. If W"(p~)r3 A-----r we proceed to the next foliation ~"(P~-I). If W~(pl) r3A. o, but W"(pi) nW'(pk) = o for i<k<j, we can proceed as before since WU(p~) r3 A is compact and disjoint from W~(pk) t3 A. Finally, let W~'(p~) n W'(p,) . o for some i < n < j. The vector field X is already defined in a neighborhood V of W"(pk) in A for i < k < j. Since .~'~'(p~) is compatible with ~'~(Pk), X is tangent to the leaves of o~-~(pi) in V. But W~(p~) n (A -- V) is compact, so we are again reduced to the previous case. Thus we can construct X as desired in a neighbor- hood of A in the center stable manifold. Now we take the integral curves of X and their positive iterates by r By the ),-lemma [24], [3o], this partial foliation extends to center manifolds and together they form a center foliation as asserted in (a). Parts (b) and (c) follow as in Proposition (3. ~) for the hyperbolic periodic points and also as in Proposi- tion (~. 3) for the saddle-node. In this last case, we raise the strong unstable foliation from the center foliation we have just constructed. The proof of Proposition (3.3) is complete. Remark. -- The center foliation ~~ is not a " classical " foliation in two ways (1) as usual in the present work, we only required that the leaves of ,~-~, which are G ~- 1 with k > 2, should vary continuously in the C ~- 1 topology. We did not demand the projection along the leaves into a transverse section (like a leaf of the strong stable foliation) to be differentiable, (2) ,~'~ has singularities, as shown in the picture FIo, xl 62 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 3 Now let us consider the situation of a saddle-node p~ = p~ +j, for Vb which is critical or lies in a cycle and j' > I. In the case the saddle-node is bicritical or it is part of a cycle, we wish to prove the stability of the family { q~ } for V~ _< b (left stability). When the saddle-node is critical but not bicritical and there is no cycle, we wish to prove that { q~} is midly stable for [z < ~1, some ~1 > b and near b. Again, since %, is Morse- Smalefor b< tz< Vtl, some [zl>b , { r } is stable in this range of the parameter. This follows from [25] or the next section. Thus, it is certainly enough to show that {?~} is left stable at b. In conclusion, we can treat these three cases in the same way. As before, we need to construct a compatible system of unstable foliations for { q~}, F <__ b. Proposition (3.4). ~ If % has a saddle-node pj =pj+j,, j' > z, which is critical or, has a cycle, then there exists a compatible system of unstable foliations for V. <_ b. The leaves of ~"(Pi+J',~,) are one dimension lower than those of pj,~. Proof. ~ Similar to that of Proposition (3.2), using the unfolding of the saddle-node as in Proposition (2.3). In this case the leaves of ~"(p~, t~) cover a neighborhood of W~'(p~, ~) --{pj,~,} near pj,~ where W*'(pj,~) is the strong stable manifold of pj,~. The leaves of ~-"(p~,~) intersected with the center stable manifold W~,' are repre- sented by horizontal lines in the pictures below. The intersections of the leaves of ~"(pj+j,,~) with W~ ~ are represented by points. "x,,x" \~'K f,- / I / / i It I , /\ /\ VN W cs 3,, ~ <b vn w? Fio. x2 Notice that the foliations ~z-,,(pj,~) and ~"(pj+j,,~) can be constructed to be compatible in a neighborhood V of WS~(p~) --p~ in M � [~1, b] for some [zl< b. Proposition (3- 5). -- If @b has a flip point pj,, then (a) there is a compatible center foliation ~ defined in a neighborhood of the center stable manifold of rb at (p~, b); 63 S. NEWHOUSE, J. PALIS AND F, TAKENS (b) there is a compatible system of unstable foliations ~"(pi) for ~, ~ ~ [~l, ~] for some ~l < b and g~ 2> b; the foliation we consider at the flip pj is a strong unstable one, ~; (c) the leaves of ~"' through a leaf of ~" form a C 1 submanifold. Proof. -- The proof is entirely similar to that of Proposition (3.3)- We finish this section by summing up in the next theorem the results we have obtained on compatible systems of unstable (strong unstable) foliations. Theorem (3.6). -- Let b be the bifurcation point of the family {%} in ~. Let @ : M � I -,'. M be defined by r ~) = (%,(x), ~). Then, (I) if% has either a noncritical saddle-node and no cycles or a flip, there exists a compatible system of unstable foliations for r or {%}, ~ < [z i for some ~l near b and ~l 2> b. The unstable foliation at the saddle-node or the flip is a strong unstable one, (2) if % has a saddle-node or a Hopf periodic orbit, there exists a compatible system of unstable foliationsfor @ or {9~}, ~ <__ b. 4- Stability We culminate this chapter by showing the stability of the families of diffeomor- phisms {%,} in ~. Such families were introduced in Chapter III, studied there and in the previous sections of the present chapter. A family { %} in M is stable, mildly or left stable according to the structure of the diffeomorphism %, where b is the first bifhr- cation point of {~}. More specifically, the stability depends on the structure of the non-hyperbolic orbit of %. We stress again that the transversality of the stable and unstable manifolds of the periodic orbits is necessary even for left stability, as was proved in Chapter III. Our results on stability will follow rather naturally and in a unified way from the existence of compatible systems of (strong) unstable tubular fatuities or foliations. Fitting together these foliations provides a kind of global coordinate system from which the topological conjugacies can be constructed. We define the conjugacies inductively on the stable manifolds of the periodic orbits, these orbits being totally ordered as in section 3. We will make use of the following two known results. The first one states that is open in the set ~ of C ~~ one-parameter families of C ~ diffeomorphisms of M with the C ~ topology. It corresponds to Theorem (2.5) of [i9] and Theorem (3. i) of [20]. The second result is an easy extension (a parametrized version) of the Isotopy Extension Theorem [~3]. Theorem (4. x). ~ Let {%} e ~ have the first bifurcation point b. There exists a neighborhood U of {~?~} such that if {~} ~ U has first bifurcation point ~, then {~,} e ~ and % and "~ have the same elementary bifurcation. Moreover, there is an order preserving one to one correspondence between the periodic orbits of % and "~. fi4 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 5 Let N be a C' compact manifold, r ~ x, and A an open subset of R ~. Let M be a C | manifold with dimM~dimN. We indicate by C](NxA, M� the set of C * mappings f: N � A -+ M � A such that n ---- ~'f, endowed with the C * topo- logy, ~ <k<r. Here, n and 7:' denote the natural projections n:NxA-+A, ~':M� Let Diff~(MxA) be the set of C *diffeomorphisms @ of MxA such that ~' -~ n'@, again with the C ~ topology. Theorem (4.~,). --Let i e C](N � A, M � A) be an imbedding and A' a compact subset of A. Givenneighborhoods U of i(N � A) in M � A andV of theidentity in Diff~(M � A), there exists a neighborhood W of i in C](N x A, M � A) such that for each j ~ W there exists , e V satisfying ~i =j restricted to N � A' and ?(x) -= x for all x r U. Now we prove the main theorem of the present chapter. Theorem ('t. 3). -- Let { ?~} ~ ~ with first bifurcation point b. Then I) if % has a flip or a noncritical saddle-node with no cycles, {~} is stable, 2) if ?b has a saddle-node which is not bicritical and has no cycles, {q~} is mildly stable, 3) if ?~ has a Hopf periodic orbit or a saddle-node which is bicritical or has a cycle then {@~} is left stable. Proof. -- First, we observe that the statement in part (2) can be proved as in part (3). In fact, in case (2) we have that {q~} is Morse-Smale for b < ~ ~ ~tt, some [z t > b, and so it is stable in this range of the parameter. This last fact is an easy consequence of the proof that 1Vforse-Smale diffeomorphisms are stable. By the same reason, in all cases it is enough to consider the stability of{ ~%} in ~ for [z near the first bifurcation point. Let Pl,~, ~ Pz,~ ~ ... ~ Pt,~ be a total ordering of the periodic orbits of ?~ for < b. As in section 3, this ordering can be naturally extended for [z = b and even for [z > b when 9b is as in parts (t) and (2) of the statement. By Theorem (3.6), we can take a compatible system of (strong) unstable foliations ~'u(pl) , ~u(pz), ..., ~(Pt) defined in 1V[ � [~t, ~z~] or 1V[ � [~zl, b], for some ~1 < b and g~ ~ b according to case (I) or (3). In case (I), the foliation is a strong unstable one at the flip or the saddle- node. Recall that this strong unstable foliation is constructed from a center foliation .,~-~ in the center stable manifold of@ at theflip or the saddle-node, where 9 : IV[ � I -+ M � I is defined by r ~) = (?~(x), ~z). By Theorem (4. I), for a nearby family {~'~} with first bifurcation point b, we can consider a corresponding compatible system of (strong) unstable foliations and a compatible center foliation in case (i). Also, given a repara- metrization (a homeomorphism) p : I ~ I near the identity and p(b) = ~, there is an order preserving continuous correspondence between the periodic orbits p~,~ and ~.p(~,) of q~, and ~p(~,), for ~ ~ [V~, ~] or V~ ~ [~z~, b]. This defines the conjugacy on the periodic orbits. In the construction of a global conjugacy we can take this repara- metrization quite arbitrarily, except in the case of a noncritical saddle-node with no 9 66 S. NEWHOUSE, J. PALIS AND F. TAKENS cycles. In this situation we must choose p as in Chapter II, so that we have a (continuous) conjugacy between (I) and @ restricted to center manifolds at the saddle-nodes. Our global conjugacy H will be constructed on M x [~l, ~] or iV[ � [~ti, b] inductively on the stable manifolds of the periodic orbits and, in case (i), at the center stable manifold of (I) at the saddle-node or the flip. Since these stable manifolds cover all of iV[x{~t} for each ~z~[~i,g~] or Ez~[~l,b], H will be defined on all of MX [~zx,~] or Mx [~t i,b] and maps onto 1Vfx [P(~i),P(~)] or MX [p(Ezx),b']. We fix fundamental domains A~ for W"(p~) or WCS(pk) with exterior boundaries transverse to all WU(pi), i < k ~ t. The conjugacy H will be constructed with the following properties: a) H(Mx{A) =Mx{p(~)}, ~in MX [~1,~] or in MX [~,b], b) it sends leaves of ~-"(pi) into leaves of ~r,(~), I < i < t, c) it is differentiable along each leaf of o~-"(pl) off the stable manifold of (I) atpl, I < i < g, d) if W"(pl) n W*(pk) ~e o, i < k, then for each leaf F of ~-"(pl), in a neighborhood of F n Ak, H is C a near the inclusion map. As the first step H takes the sources of{q~} onto the sources of {~'~(~1}. Suppose H has been constructed on the stable manifolds of @ at Pi > P~ > -.- > P,. Notice that the space of leaves of ~"(pi) is parametrized by the intersection of the leaves with W'(pi) or W~(p~). Thus, in particular, H defines a map from the space of leaves of ~'~(pl) onto the space of leaves of ~"(~) for I < i < n: F ~ ~"(p,) is associated to F ~ o*-"(~) if H(F n W*(p~)) F c~ WS(~) Let us now consider W~(P.+i), W (P.+z) ifP.+i is either hyperbolic, a Hopf orbit or a saddle-node which is critical or has a cycle. Let Ab-~A.+in(Mx{b}) and Ag=A.+1n(Mx{~)). If W"(p.) nAb4=O, we choose H as a diffeomorphism near the inclusion of a neighborhood of Wt'(pn) n 0e.Ab in W~(p.) onto a neighborhood of W~(~.) n 0o.Ag in WU(ff.). Such a ditTeomorphism exists because W"(p.) n A b is close to W"(~) n A~ for {9~} near {~,}. Near the interior boundary of Ab, H is defined by H% = ~g H. From Theorem (4. ~) this partial diffeomorphism can be extended to all of WU(p.) n A b. To extend H to the leaves of ~'u(p.) near A~+ l, ~ <_ ~ ~ b, we proceed in an entirely similar way using Theo- rem (4. ~), since we already know which leafof~(~) is associated to a given leaf of #-'*(p,,). This completes the construction of the conjugacy on the intersection of the leaves of ~'u(p.) with the fundamental domain A.+ 1. Next, we take ~(Pn-1) and suppose that W'*(p~_l) n A b 4: o. There are two cases to consider. If WU(p,,_ i) c~ W'(pn) = o, then W~(p._ i) n A b is compact and disjoint from WU(p~) ~ A b. Thus, the cons- truction of the conjugacy on the leaves of .~-u(P.-1) restricted to A.+a is the same as on the leaves of ~"(p.). So, assume that W~(p._ i) n W'(p.) 4: o. Since ~-"(p,~_ i) and 9 ~'u(Pn) are compatible, H is already defined on the leaves of~'~(i0n_ x) in a neighborhood V of W"(p.) c~A n inA.+i. But B=W"(p._l) n(A b-V) is compact. So, we can extend H to B using Theorem (4.2) and also to the leaves of ~*"(p._ i) in a neighborhood of BinA.+l. We repeat the argument to all ~(p~), I <i<n--2. IfP.+iis a 66 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 7 Hopf orbit or a critical saddle-node with no cycle, then H is defined on A.+ i satisfying H?~, = ~p(~,)H or H 9 = ~H, ~1 ~ ~ ~ b or ~1 ~ ~ ~ ~ with ~1 < b and b < ~,. The same equation allows us to define H on all of the stable manifold of @ at p.+i. If the saddle-node occurs in a cycle so that P.+x = P.+ l+j' for some j'~ I, then we have only defined H on a neighborhood V of W"(p.+l,b)--{P.+a,b} in W~(p,,+l). We continue as before forp.+l+~with j <j'. Whenwecome top.+l+~, , the conjugacy H as already been defined on the part V of W'(p. + 1+4'). We continue the process as above obtaining a conjugacy Ha defined on (k<.~l+j' W"(Pk))n W'(p.+l+~, ) such that H i ~-H off a neighborhood of W'(p.+l+i, ) --V in W'(P.+l+j,). Thus, H can be defined on all of WS(p.+l). Now we consider the case where P.+l is a flip or a non-critical saddle-node with no cycles. Here, H will be defined in the center stable manifold of @ at p.+ 1 as the "product " of two partial conjugacies. One of them, which we call H e, is defined at the center manifold of @ at p. + x, as constructed in section 3 of Chapter II. It corresponds to a conjugacy on the space of leaves of the strong stable foliations ~"(p.+ 1), ~r.+ aJ. The other, which we call H', is to be defined on the space of leaves of the center folia- tions ~"(P.+i), ~'~(~+ 1). However, since the center foliations may have several leaves going through the periodic orbits P.+I, ~+i, we cannot express the stable manifolds of such orbits exactly as the product of these two foliations. That is the reason we will make a slight modification of the center foliation ~-'(P.+I) already constructed. The second partial conjugacy H' will then send leaves of ~-"(p.+ i) onto leaves of the modified center foliation for ~. + 1. To do this, we first consider a continuous family of leaves { F~, }, _ _ ~r.+ l, ~), such that F~, meets all the leaves of the center foliation ~'~(P.+i, ~). We also choose a family of disks {D~}, D~,CF~,, such that the exterior boundary 0~. D~, of D~ is transverse to the unstable manifolds W"(p~, ~) for all I <i<n and each ~e[~l,~]. We now form a fundamental domain A= UA~ for the center stable manifold of/,. + 1 as follows. For each ~, the exterior boundary 0~. A~ is made of the disk D~, and a " cylinder " formed with the leaves of o~-"(p.+ i, ~) through the points of 0e. D~. Notice that H" determines a corresponding family { F~(~)} of leaves of ~z-~t~'~r.+ lJ~. We then take a family of disks {Dp(~,)}, D~(~,)C F~(~,) for each ~, and construct a similar fundamental domain A = U .~(~) for the center stable manifold of~+a. We now define H" : U D~ -+ U Do(~,) as follows. Of course, we want it to be compatible with the unstable foliations ~-"(pl) and the conjugacies already defined on the stable manifolds W'(pi), I < i < n. Near the exterior boundaries of D~, D~(~) we proceed by induction on the indices i = n, n -- z, ..., ~. We start with a diffeo- morphism H":V ~V near the inclusion map, where V is a neighborhood of W'*(p~) ~ 0exD ~ in W"(pn) c~ F~ and V is a neighborhood of W"(~) ~ 0exD0(~) in W"(~.) c~ F0(~)" Similarly for the intersections of the leaves of ~"(p.) with 0o.D~ which are mapped onto the intersections of the corresponding leaves of o~-"(~.) with 0,. D0(~)" Notice that the correspondence between the leaves of ~"(p.) and those of ~'~(~.) is 67 68 S. NEWHOUSE, J. PALIS AND F. TAKENS determined by H:WS(p,) -+ Ws(p,) already defined. The extension to the leaves of ~"(P,-1), 9 9 -, ~"(Pl) is done in the same way as in the previous case. In order to define H ~: U D~ --> U D~(~), we consider the annuli A~CD~ and ~(~)CDp(~), each ~z E [Or1, W], obtained as follows. The exterior boundary of A~, is 0e.D ~ and its interior boundary is the projection of ?~,(Oe, D~,) into D~ via the leaves of ~-c(p,+l, V~). Similarly for "~C~)" Notice that H~:Oe~A~,-~ 00.Ao/~) constructed above induces a map H* : 0i.A ~ -+ Oi, A~(~). In fact, for x ~ 0i.A ~ let g, be the leaf of ~'*(P,+I, ~) through x and let y be the intersection of ~-t(t,) with 0e.A~. If 7 is the leaf of ~-"(~+ a, p(~z)) through H*(y), we set H"(x) = z, where z is the intersection of ~p(~)(7) with Oi, A~c~). We now extend H' first to all of A; using the Isotopy Extension Theorem as before. Finally, we can extend H 8 to all of D~ sending leaves of #z-,,(p~, ~z) to leaves of ~-"(~, p(tz)), i < i < n. Again, this can be done as before since W"(p~, t~) and W (p~, O(~)) are transverse to D~ and D0(~) , respectively. At this point, we would like to define the conjugacy H : W**(p,+l) ~ W"*(p,+I) as the " product" of H* and H' using the strong stable and the center foliations. To do this we have to modify the center foliation ~'*(P,+I) in (P,+x). For each V~ ~ [tzl, Vt,], it is enough to do so in the region bounded by Dp(~)- ~l~), ~(~)(Dp(~,)) and the cylinder formed with the leaves of ~*(~+~) through points of 0i,A~(~, ). Let ~q :Do(~)- be the homeomorphism defined by ~q(x) =y, where (H*)-~(x) and q~(H')-t~]-~](y) belong to the same leaf of ~*(p,+x, ~). Let ?,: D0(~)- A~(~)~(~,)(D0(~)) be defined by ~,(x) = z, where x and z belong to the same leafofo~-*(~+x, p(~z)). If 2, = ~ then no modification of ~-*(~+~, O(~)) is needed. So let us change o~'0(~,+~, p(iz)) to get the second map to be equal to the first. Notice that ~q(x) and ;~(x) belong to the same leaf of ~-"(~, p(~)) for some ~ < i < n. Moreover, ~,~-~ along such a leaf is C t near the identity map. Let X be the vector field whose integral curves are the leaves of t~a o*-*(~+~, p(tz)). We modify X near but off q~o(~)(D~(~)) so that the corresponding map ), satisfies 7,~- ~ --~ I on @"~(~)(Do(~)). Observe that the modification required for X along W (p,, p(~z)) is well known and it can be performed in a parametrized way along the leaves of ~"(~,, p(~)). Using the fact that the foliations ~"(~, 9(~)) are compatible, we proceed by downward induction on the indices I < i < n. It is clear that the new center foliation coincides with the previous one on the boundary OA = O OAo(~, I of the fundamental domain A. Once it is defined on A, it can be defined on all o ,, ~/,, + x) simply through iterations by ~, where ~(x, tz) = (~"~(x), ~) as before. Now we can W e*t~" '~ using the center foliation, the strong stable define H from W=(p,~+~) onto ~v,+~ foliation and the conjugacies H" and H ~ on their spaces of leaves. H is clearly one to one, onto and also continuous since these foliations and the maps H 8, H ~ are continuous. Moreover, H is differentiable along the leaves of #-"(pl), I < i < n, in W**(p,+l). In fact, ~c was constructed in section 3 to be differentiable along these leaves; the same is true with the modified #z-* and the map H ~ constructed above. Being codimension one, the foliation 9~'** is C t (see section ~, Chapter II). Finally, the differentiability A~(~,)-*~"~(~,)(D~(~,)) BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 9 of H a off the periodic points is in section 3 and 4 of Chapter II. Together these facts imply our statement. Thus, the proof of the induction step is finished. Our map H is defined on M � [~1, ~] or M � [~1, b] onto M � [P(~I), P(~,)] or M � [P(~I), ~], it is clearly one to one and satisfies the conjugacy equation H~ = ~'o(~)H. Let us prove that H is continuous. We have to show that H is continuous at the stable (center stable) manifolds of the periodic orbits of {q~}. Here we indicate by W 8 both the stable and center stable manifolds; also, let 11 indicate either [~1, ~] or [~1, b]. By construction, H is continuous along these manifolds and, in particular, along the stable manifolds of the sinks. By induction we may assume that H 9 W ~ W is a homeomorphism, where W and W are the union of Wa(pk,~,) and Wa(pk,~(~,)) for i< k <_/ and ~ EI 1. Let us now show that H is also continuous at Wa(p~,~,). Consider a sequence (x,, ~,) ~ (x, ~) ~ Wa(p~,~), x, ~ M and ~, E 11. Since H restricted to Wa(p~,~,) is continuous, we may assume, via a subsequence, that (x,, ~,) ~ W for all n. Let F,, F be the leaves of~"(p~) containing (x,, ~,), (x, ~) and F,, F the leaves of ~"(~) contai- ning H(x,, ~,) and H(x, ~), respectively. We have that F, -+ F and since H is conti- nuous restricted to the stable manifolds, we have F, -+ F. Thus, it is enough to show that the sequence H(x., ~,) accumulates on W (p~,~(~)). In fact, H(x,, p~,,) cannot accumulate on W because H : W -+ W is a homeomorphism. Also, H(x,, ~,) cannot accumulate on the union Z of W (p~,~(~)), I <__j < i, because F, and Z are far apart. This proves our assertion and so H is continuous on all of M � I~. This finishes the proof of the theorem. We now complete the proof of our second main theorem stated in the introduction, the first part of which was done in section ~ of Chapter III. Let us denote by ~ C ~ the set of arcs { ~, } such that the limit set of each ~, has finitely many orbits, ~ ~ I = [0, 1]. We also denote by 5~CCg the set of arcs {~?~,} such that there are only finitely many bifurcation values for { @~,} say bx, ..., b, in (o, I) and for each ~ < i < s, %; has the following properties: -- all stable, strong stable, unstable and strong unstable manifolds intersect transversally -- %i has no cycles and has exactly one non-hyperbolic periodic orbit, which is either a flip or a non-critical saddle-node; this non-hyperbolic orbit unfolds generically. As we mentioned before, it turns out that for arcs { ~ } ~ 5r ~ is a Morse-Smale diffeomorphism if ~ is not a bifurcation value. Theorem (4.4). -- The arcs in 5P are stable. Proof. -- As in Theorem (4- i), it follows from [i9] , [2o] that 5P is an open subset of~. Thus, if {~} ~ 5r and has bifurcation values bl, ..., b a in (o, I), then a nearby t t t arc { @~,} also belongs to 5r and has nearby bifurcation values bl, ..., b, in (o, i). More- over, the non-hyperbolic periodic orbits of %1 and ~'b} are both saddle-nodes or flips. To produce a conjugacy between {q~,} ~ 5r and a nearby arc {~%}, we first assume that 69 7 ~ s. NEWHOUSE, J. PALIS AND F. TAKENS 9~ = r for ~ ~ I -- U, where U is a small open subinterval of I containing at most one bifurcation value. In this case, by Theorem (3-6) we can construct unstable tubular families or foliations for { r } and { q~ } and ~ ~ V, where V is art open subinterval of I containing the closure U of U. Moreover, using the Isotopy Extension Theorem (4-2), we may construct these unstable foliations to be the same for { r } and { q~'~, } if ~ e V -- W, where W is an open subinterval of I such that U C W and W C V. Again using the Isotopy Extension Theorem, the construction of the conjugacy { h~, } between { q~ } and { q~'~, } for ~ ~V, as performed in Theorem (4.3), can be done so that h~, is the identity map on IV[ for ~ ~ V -- W. Since r = r for ~ e I -- U, we can extend this conjugacy to all of I by defining it to be the identity for ~ ~ I -- V. Finally, let {Ui}, I ~ i < n, be a covering of I by small subintervals, each containing at most one bifurcation value. It is immediate that we can decompose any small perturbation of{ q~,} into perturbations, each with support in one of the subintervals U~. So the construction of the conjugacy between { r and a nearby arc is reduced to the previous case. The proof of theorem is complete. REFERENCES [I] A. A. A~mRoNov, E. A. L~.o~rrovlc~, I. I. Go,oN, A. G. MAmR, Theory of bifurcations of dynamic systems on a plane, Isr. Program Scientific Transl., Jerusalem, x97x. [2] L. BLOCK, J. FV.ANmZ, Existence of periodic points of maps of S I, Invent. Math., 22 (1973), 69-73. [3] Th. BR6cmm, L. C. LANDER, Differentiable germs and catastrophes, London Math. Soe. Lecture Notes, 17, Cambridge Univ. Press, 1975. [4] P. BaUNOVSKt, On one-parameter families of diffeomorphisms, I and II, Comment. Mat. Univ. Carolinae, 11 (3) (x97o), 559-582; 12 (4) (x97t), 765"784 9 [5] P. BRtrNOVSKI, Generic properties of the rotation number of one-parameter diffeomorphisn-Ls of the circle, Czech. Math. f., P~ (I974), 74"9o. [6] A. D,~joY, Les trajectoires ~ la surface du tore, C. r. Acad. Sei., 9.23 (t946), 5-8. [7] J. GUCraZNH~IMER, One-parameter families of vector fields on two-manifolds: another nondensity theorem, Dynamical Systems, Academic Press, x973, I i x-128. [8] P. HART~a'% On local homeomorphlsms of Euclidean spaces, Bol. Soe. Mat. Mexicana, 5 (2) (I96O), 22o-24~. [9] M. H,~mt~a~, Sur la conjugaison diffdrentiable des diff~morphismes du cercle ~t des rotations, Publ. Math. I.H.E.S., 49 (I979) , 5-234. [xo] M. HEma~x, Mesure de Lebesgue et nombre de rotation, Geometry and Topology, Springer Lecture Notes in Math., 597 (i977) , 27x-293. [Ii] M. W. HmscH, C. C. PuoH, Stable manifolds and hyperbolic sets, Proc. Syrup. Pure Math. A.M.S., 14 (197o), x33-x64. [I2l M. W. HImCH, C. 0. PUGH, M. Smm, Invariant manifolds, Springer Lecture Notes in Math., 583 (x977). [x3] I. Ktmr,~, Contributions ~ la th~orie des champs g~n~riques, Contributions to differential equations, vol. 2 (x963), 475-484 and vol. 3 (I964), 4xt-42~ [i4] D. C. LEwxs, Formal power series transformations, Duke Math. f., 5 (x939) , 794-8o5. [15] L P. MALTA, Hyperbolic Birldaoff centers, Trans. A.M.S., 262 (x98o), x8x-193; announced in An. Aead. Brasil. Gi~ias, 51 (1979), (1), 27-~9. [t6] A. Mam~x~o, There are no new Anosov diffeomorphisms on toil, Amer. f. Math., 96 (x974), 422-4~9. [x7] J. E. M~o.~, M. MeC~x~, The Hopf bifurcation and its applications, Appl. Math. Sd., 19, Springer- Verlag, ~976. [tS] W. C. de M_.~LO, Moduli and stability of two-dimensional diffeomorphlsms, Topology, 19 (~98o), 9-2~. 70 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 7 t [18a] W. de MLO, J. PALxS and S. VAN Sami~, Characterizing diffeomorphisms with modulus of stability one, Dynamical SysWms and Turbulence, Warwick 198o , Springer Lecture Notes in Math., 898 (I981), 266-286. [19] S. NEW~OtrSE, J. P~as, Bifurcations of Morse-Smale dynamical systems, Dynamical Systems, Academic Press, 1973, 3o3-366. [20] S. NEwaousE, J. PALls, Cycles and bifurcation theory, Ast~risque, 31 (1976), 43-x4o. [2x] S. NEWHOUSE, J. Pro.Is, F. TAKeNS, Stable arcs of diffeomorphisms, Bull. A.M.S., 82 (1976), 499-5o2. [22] S. NEwHotrse, The abundance of wild hyperbolic sets and non-smooth stable sets, Publ. Math. KH.E.S., 50 (1979), Ioi-152. [23] R. P.~J~lS, Local triviality of the restriction map for embedding, s, Comm. Math. Helvet., 84 (I96o), 3o5-312. [24] J. PAXaS, On Morse-Smale dynamical systems, Topology, 8 (1969), 385-4o5. [25] J. Proem, S. SMM.~, Structural stability theorems, Pror Syrap. Pure Math. A.M.S., 14 (t97o), 223-232. [26] J. PALLS, F. T~NS, Topological equivalence of normally hyperbolic dynamical systems, Topology, 18 (i977) , 335-345. [26a] J. P~ts and F. T~.eNS, Stability of parametrized families of gradient veaorfields, Preprint IMPA, to appear. [27] J. PALm, A differentiable invaxiant of topological conjugacies and moduli of stability, Asttrisque, 51 (x978), 335-346. [28] J. PALLS, Moduli of stability and bifurcation theory, Proc. Int. Congres of Math. Helslnki 0978), 835-839. [29] H. Pomea~, (Euvres completes, t. l, Gauthier-Villars, x952 , t37-x58. [3 o] C. C. Puoa, M. SHtm, Linearization of normally hyperbolic diffeomorphisms and flows, ImJent. Math., 10 (197o), 187-198. [3 l] C. RomNso~, Global structural stability of a saddle-node bifurcation, Trans. A.M.S., 236 (x978), 155-172. [32] D. RtmLLE, F. T~ENS, On the nature of turbulence, Comm. Math. Phys., 20 (x97x), t67-x92; A note concerning our paper on the nature of turbulence, Comm. Math. Phys., 28 (197x), 343-344- [33] S. SM~, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa, 18 (I963), 97-I x6. [34] S. S~Le, Differentiable dynamical systems, Bull. A.M.S., '/8 (t967), 747-8x7. [35] J. SOTOMAYOa, Generic one parameter families of vector fields, Publ. Math. LH.E.S., 43 (1974) , 4-56. [36] J. SOTOMAYOR, Generic bifurcations of dynamical systems, Dynamical Systems, Academic Press t973, 549-56o. [37] S. J'. VAN STmEN, Center manifolds are not C ~~ Math. Z., 168 (I979), x43-145. [38] F. T~:zNs, Partially hyperbolic fixed points, Topology, 10 (197x), x33-I47. [39] F. T~a~Ns, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier, 28 (t973), (2), 163-t95. [4 o] F. T~NS, Forced oscillations and bifurcations, Applications of Global Analysis I, Comm. of the Math. Inst., R.U. Utrecht (Holland), t974. [4 l] F. TAKE.~S, Global phenomena in bifurcations of dynamlcal systems with simple recurrence, fber. d. Dr. Math.- Vereln, 81 (a979), 87-96. [4 ~]R. W~a~MS, The D.A. map of Smale and structural stability, Pror Symp. Pure Math. A.M.S., 1~ (197o),329-334. Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514, U.S.A. I.M.P.A., Rua L. de CamiSes 68, Rio de Janeiro, Brazil. Department of Mathematics, University of Groningen, Holland. Manuscrit re~u le 22 avril i98t. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Bifurcations and stability of families of diffeomorphisms

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Springer Journals
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Copyright © 1983 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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0073-8301
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1618-1913
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10.1007/BF02698773
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Abstract

We consider one parameter families or arcs of diffeomorphisms. For families starting with Morse-Smale diffeomorphisms we characterize various types of (structural) stability at or near the first bifurcation point. We also give a complete description of the stable arcs of diffeomorphisms whose limit sets consist of finitely many orbits. Uni- versal models for the local unfoldings of the bifurcating periodic orbits (especially saddle- nodes) are established, as well as several results on the global dynamical structure of the bifurcating diffeomorphisms. Moduli of stability related to saddle-connections are introduced. CONTENTS CI~R I. -- Introduction ................................................................. 7 CHAPTER II. -- Local description of the elementary bifurcations ............................ 13 I. Introduction ......................................................................... 13 2. Center manifolds and periodic elementary bifurcations ................................... 13 3- The saddle-node ..................................................................... 15 4. The fllp ............................................................................. 2x 5. The Hopf point ...................................................................... 22 6. Quasi-transversal intersections .......................................................... ~ CHAPaXR III. -- Necessary conditions for stability of ares .................................. 26 i. The modulus condition (quasi-transversal intersection) .................................... 26 2. Necessary conditiom for mild stability and stability ...................................... 37 3. Endomorphisrm of the circle .......................................................... 43 4. The saddle-node with l-cycle in dimension 2 ........................................... 48 5. On the rigidity of the unfolding of the saddle-node ...................................... 5x 5 S. NEWHOUSF_,, J. PALIS AND F. TAKENS CHaR IV. -- Global stability ............................................................ 55 I. Introduction ......................................................................... 55 2. Local tubular families ................................................................ 55 3. Global tubular families ............................................................... 59 4. Stability ............................................................................ 64 This research was carried out mainly at the Instituto de MatemCttica Pura e Aplicada, Rio de Janeiro. The first two authors were also supported by an agreement between GNPq (Brazil) and NSF (USA). I. -- INTRODUCTION The study of the geometric structure of the orbits of dynamical systems (differential equations, flows, vector fields, or diffeomorpkisms) defined on a manifold has been considered in many works since Poincar~ and Liapunov. Two diffeomorphismsfandf' are said to have the same geometric structure if they axe topologically conjugate, i.e. if there is a homeomorphism h from the domain off to that off' such that hf =f' h. Two flows or vector fields, are called topologically equivalent if there is a homeomorpkism sending orbits of one system to orbits of the other; if, in addition, the homeomorphism preserves the flow parameter, we again say the systems are conjugate. In general terms, we aim at the classification of dynamical systems under conjugacy or topological equi- valence. Since, however, much pathological behavior can occur, we must restrict ourselves to interesting special classes of systems. We shall be concerned here with systems having only mild recurrence; in particular, we shall frequently assume that their limit sets consist of only finitely many orbits. The space of differentiable systems of class C r, r > I, has a natural topology given by uniform convergence of the first r derivatives. This is called the C' topology. Given any equivalence relation E on the set of dynamical systems, one can define systems to be E-stable if they lie in the interior of their E-equivalence classes. When topological equivalence is used for E, an E-stable system is called structurally stable (or just stable). The stable diffeomorphisms and vector fields whose limit sets have finitely many orbits coincide with the Morse-Smale ones [24]. In fact, in this case the Birkhoff center [I5] is finite because non-trivial recurrence implies uncountably many orbits in the limit set. Since one understands the structure of Morse-Smale systems pretty well, it is natulal to consider one-parameter families of systems starting at a Morse-Smale one and to attempt to describe the structure of the elements of such families. In the present work we shall define three natural equivalence relations on these families, and we shall characterize their stable families in terms of geometric properties. In particular, we will characterize the stable one parameter families of diffeomorphisms whose elements have only finitely many orbits in their limit sets. This corresponds to the characterization of the Morse- Smale diffeomorphisms as the ones which are stable and have finitely many orbits in their limit sets. The results can, of course, be translated to certain classes of vector fields (those with global cross-sections). In [26a] related results are obtained for families of gradient vector fields. We first present a preliminary description of our main results. Later, we shall give their precise statements. S. NEWHOUSE, J. PALIS AND F. TAKENS In [I9] , [20], one studied how a one-parameter family of diffeomorphisms starting at a Morse-Smale one ceases to be stable (i.e. goes through a bifurcation) when the parameter evolves. For a generic family, the description given there is complete assuming that the diffeomorphism at the frst bifurcation point has its limit set made of a finite number of orbits. It turns out that these orbits are all periodic except at most one. If the periodic orbits are all hyperbolic, then their stable and unstable manifolds meet transversally except along one orbit. In case one of the periodic orbits is not hyperbolic, this orbit must be an elementary bifurcation (a saddle-node, flip, or Hopf orbit), the other periodic orbits must be hyperbolic, and all stable and unstable manifolds meet transversally. The orbit structure of the diffeomorphism at the first bifurcation point will be basic for our results on the stability of these arcs. Throughout this paper, M denotes a compact C ~~ manifold without boundary, and Diff(M) denotes the set of C | diffeomorphisms of M. We let 9 ~ ---- 9~(M) be the space of C ~~ arcs ofdiffeomorphisms on M. That is, if I is the unit interval, then ~(M) consists ofG ~176 mappings 9 : M � I ~ M � I, such that ~(m, ~) = (q~(m), ~) where m ~ ~,(m) is a G ~~ diffeomorphism for each Ez ~ I. Elements of ~ will also be called one-parameter families or arcs of diffeomorphisms and will frequently be denoted by { %,} or q~. We give Diff(M) and ~ the usual &o topologies. Let us consider three equivalence relations on the set ~ of one-parameter families of diffeomorphisms. We say two families are topologically conjugate if, modulo an orien- tation preserving homeomorphism of the interval I, each element of the first family is topologically conjugate to an element of the second family, and the conjugacy varies continuously with the parameter. If conjugacies exist but do not necessarily vary conti- nuously, we say the families are mildly conjugate. Finally, if the elements of the families are topologically conjugate up to and including their first bifurcation points, we say they are left conjugate. The interiors of the equivalence classes of the preceding equivalence relations define, respectively, stable, mildly stable, and left stable arcs of diffeomorphisms. For arcs (~} such that the limit set of r consists of finitely many orbits for each ~z, we will give necessary and sufficient conditions for stability, and we will geometrically characterize left stability. For mild stability, our characterization is complete except for one case which we present as an interesting open question. In aU cases, we exhibit necessary and sufficient conditions in terms of the orbit structures at the bifurcation points of the arcs. For arcs beginning at Morse-Smale diffeomorphisms we will first study the different types of stability for an interval in I containing the first bifurcation point in its interior. Let us describe the results. To begin with, if, at the first bifurcation point, some stable and unstable manifolds meet non-transversally, then the arc is not even left stable. In fact, in this case we have at least a one-parameter family of different equivalence classes of arcs near the initial one. This corresponds to the existence of the modulus of stability as discussed in [27], [28]. In the other possible cases, when there is a saddle- node, a flip or a Hopf periodic orbit, the arc is left stable. When the arc goes through BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS a Hopf orbit it is never mildly stable; this is due to the appearance of invariant circles with irrational rotations. On the other hand, we always get stability for arcs going through flip orbits. The case of a saddle-node orbit deserves a special discussion. First of all, it is much harder than in the flip case to prove the existence up to conjugacy of a universal model for its local unfolding. This is done in Chapter II. On the other hand, the restriction of such a conjugacy to the center manifold is surprisingly rigid. We discuss several applications of this fact in section 5 of Chapter III. Moreover, the strong stable and strong unstable foliations of the stable and unstable manifolds must be preserved by a conjugacy between two arcs going through saddle-nodes. Thus, criticaUity or tangency of the invariant (stable, unstable) manifolds of other periodic orbits with respect to one of these foliations implies that the arc is not stable. CriticaUity with respect to both foliations impfies that the arc is not even mildly stable. Another crucial factor is the existence of cycles for the periodic orbits. When the arc goes through a saddle-node which is critical but not bicritical and has no cycles, then this arc is mildly stable but not stable. If there is a cycle, the arc is not stable and, we believe, it is not even mildly stable. We are able to prove this last statement for cycles of length bigger than one and for one- cycles when the saddle-node is normally attracting or repelling. This follows from the appearance of a non-transversal homoclmic orbit, which implies a non zero modulus of stability. To prove the existence of such a homoclinic orbit, we reduce the question to one-parameter families of endomorphisms of the circle and introduce a generalized notion of rotation number. Necessary conditions for the types of stability mentioned above are also established in Chapter III. The woof that, under these conditions, the arcs are stable, mildly or left stable is performed in the last chapter. There we use a suitable version of tubular families or foliations, some of them with singularities. Our constructions also provide a more elegant woof of the stability of Morse-Smale diffeo- morphisms originally established in [24], [25]. Stability for arcs containing saddle-nodes was generalized by Robinson [31 ] to certain families starting at Axiom A diffeomorphisms. Many of these results were announced in [2I]; however, we mistakenly claimed to have characterized mild stability. As we mentioned above, it remains to prove that certain /-cycle cases are not mildly stable. Let us now review some definitions and be more precise. For g e Diff(M), the orbits(x), of a point x ~ M, is defined as ,(x) = {g"(x) [ n ~ Z}. A point y E M is called a limit point of g if for some sequence n~ ~ Z with [ nil -~ 0% limg'~(x) =y. We denote by L(g) the closure ofthe set of these limit points. A point x ~ M is a periodic point ofg with period k if gk(x) = x and gt(x) 4= x for all o < t < k; x is hyperbolic if dgk(x) has no eigenvalues on the unit circle. The stable, unstable sets or manifolds W'(x, g), W~(x, g), of a periodic point x are defined as {y ~ M [ p(g"(y), g"(x)) -+ o for n ~ + o0} and {y c M l o(g"(Y), g"(x)) -+ o for n ~ -- oo}, 2 IO s. NEWHOUSE, J. PALIS AND F. TAKENS respectively, where p is a metric on M. Ifx is a hyperbolic periodic point of g, W'(x, g) and W"(x, g) are injectively immersed sub-manifolds of M. We say that a diffeomorphism g 9 Diff(M) with finitely many periodic orbits has a k-cycle if there is a sequence of periodic orbits ~(P0), ..., r(Pk) with ~(P0) = ~(Pk) and r(Pi+t) Cclosure (WU(~(pi)) for o < i < k, and r(p~) 4= r(pj) for o < i < j < k, and if this sequence is maximal. A diffeomorphism g is Morse,Smale if L(g) is finite and hyperbolic and if all the intersections of stable and unstable manifolds are transverse. We denote the set of Morse-Smale diffeomorphisms by MS. This set is open [24] and each g e MS is stable in the sense that any g' which is C t near g is conjugate to g [24], [25], i.e. there is a homeo- morphism h : M ~ M so that g'h = hg. For an arc {q~,} ~ with q~0 eNIS, let b = b(~) = inf{~t eI [ q~, CMS}. We always assume that b(~) < I. If {%,},{9'~,} 9 thenwesay that (h,{H~}) is aconjugacy if h : [o, I] ---> [o, I] is a homeomorphism with h(b(~)) =b(~?'), H~,:M~M is a conjugacy between q~, and 9~/~,)for all ~t in some neighborhood of [o, b(~)], and H,, depends continuously on ~t. If H~, does not necessarily depend continuously on ~t, we say that (h, {Hr,}) is a mildconjugacy, and ifH~, is only a conjugacy for ~z < b(q~), not necessarily continuous in V, then (h, {H~,}) is called a left eonjugaey. Clonjugacy, mild conjugacy and left conjugacy define equivalence relations in the set of those arcs in ~ which start in MS. An arc (~,} e ~ is called stable, mildly stable or left stable if is an interior point of its corresponding equivalence class. Now we come to the description of the class of arcs to which our results apply. Definition.- z~r is the subset of those {q~.} e~ such that I. q~o e MS; 2. b = b(q~) = inf{~t ~[o, t] I ~ r I; 3. the limit set of ~ has finitely many orbits. For the arcs of diffeomorphisms in ~ it is often useful to impose certain generic conditions on the first bifurcation. In order to describe them we need some more conditions. Let x be a fixed point ofa diffeomorphism g e Diff(M). We call x quasi-hyperbolic if one of the following three conditions holds: -- (dg)x has one eigenvalue one, the other eigenvaiues have norm different from I and there is a g-invariant curve = through x such that g J - has first but not second order contact with the identity at x; in this case, x is a saddle-node; -- (dg)x has one eigenvalue -- x, the other elgenvalues have norm different from I and there is a g-invariant curve ~ through x such that g2 [ = has second but not third order contact with the identity at x; in this case, x is a flip. -- (dg), has a pair X 4= X of eigenvalues on the unit circle, the other eigenvalues have norm different from i and there is a g-invariant surface ~ through x, tangent to the 10 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS x z generalized eigenspace of the pair 7,, ). at x and such that the 3-jet ofg at x makes g [ = an attractor or a repeller; in this case, x is a Hopf point. For periodic orbits there is an analogous definition of quasi-hyperbolicity. When {%} ~ .~g, we say that (%} is elementary at its first bifurcation value b (or % is elementary) if it fulfills one of the two following conditions: (I) all periodic points of % are hyperbolic, there is one orbit of non-transversal intersections of a stable and an unstable manifold, and all other intersections of stable and unstable manifolds are transversal; or (2) there is one quasi-hyperbolic orbit, the other periodic orbits are hyper- bolic and all intersections of stable and unstable manifolds are transverse. In case x is a flip or saddle-node of ~b we also require stable and unstable manifolds to be transversal to the strong stable and unstable manifold of x; the strong stable (resp. unstable) manifold consists of the points y such that the distance from x to q~(y) (resp. ~-~(y)) goes exponentially to zero (see also Chapter IV). If {%} ~ ~, ~b is elementary, and % has a quasi-hyperbolic periodic orbit, then there are generic conditions one may impose on the dependence upon tz at these quasi- hyperbolic periodic points. Such conditions are described in Chapter II, w 3 (for the saddle-node), w 4 (for the flip) and w 5 (for the Hopf orbit). If these conditions are satisfied we say that the quasi-hyperbolic orbit unfolds generically. Definition. -- ,~C ~t' is defined to be the set of those arcs {%} in a~ for which % is elementary and for which the quasi-hyperbolic periodic point of %, if there is any, unfolds generically. It can be shown [4], [x9], [2o], [3 6] that there is a residual subset M' in ~ such that int(~) = d ng'. We want to give necessary and sufficient conditions for arcs in g to be stable, mildly stable or left stable. For this we need to describe the notion of criticallity. Let g e Diff(M) and let x be a saddle-node ofg. Then there is a unique foliation o~ TM of W'(x, g) with smooth leaves such that the boundary of W~ g) is a leaf and such that g maps leaves to leaves; see [I 2]. ~"' is called the strong stable foliation. A corresponding foliation of WU(x, g), the strong unstable foliation, is denoted by .~-". We call x s-critical if there is some hyperbolic periodic point y of g such that W"(y, g) intersects some leaf of ~"~ non-transversally; u-critical is defined similarly. Now, x is called semi-critical if it is either s- or u-critical, x is called bi-critical if it is both s- and u-critical, and x is called non-critical if it is not semi-critical. Theorem. -- Let { ~} be an arc in ~. I. { ~} is left stable if and only if all stable and unstable manifolds of ~ intersect transversally; 2. if{%) is left stable, if the quasi-hyperbolic orbit is not a bi-critical saddle-node or a Hopf orbit and if ~b has no cycles, then { %} is mildly stable; 11 I2 S. NEWHOUSE, J. PALIS AND F. TAKENS 2'. /f{%} is mildly stable, then {%} is left stable, the quasi-hyperbolic orbit is not a bi-critical saddle-node or a Hopf orbit, % has no cycles of length greater than i, and ~b has no non-critical 1 -cycles; 3. { ~} is stable if and only if{ %,} is left stable, the quasi-hyperbolic orbit of ~b is not a semi-critical saddle-node or a Hopf orbit and ?b has no cycles. This theorem, together with the remark at the end of section 2, Chapter III, and Theorem (4.4) in Chapter IV implies the following characterization of stable arcs of diffeomorphisms with limit sets consisting of finitely many orbits. Theorem. -- Let { q~ }, Vt ~ [o, I ] be an arc of diffeomorphisms such that the limit set of each ~?~ consists of finitely many orbits, ~ ~ [o, i]. Then { ~} is stable if and only if there are onlyfinitely many bifurcation values, say bx, ..., b,, in (o, I) and for each i < i < s, ~bl has the following properties: -- all stable, strong stable, unstable, and strong unstable manifolds intersect transversally; -- ~bi has no cycles and has exactly one non-hyperbolic periodic orbit which is either a flip or a non-critical saddle-node; this non-hyperbolic periodic orbit unfolds generically. We acknowledge useful comments by several colleagues, especially W. de Melo, F. Przytycki, and C. Pugh. 12 II. -- LOCAL DESCRIPTION OF THE ELEMENTARY BIFURCATIONS I. Introduction Let M be a smooth manifold. We shall consider smooth arcs of diffeomorphisms ?~:M-+M, ~t~R, i.e. such that the corresponding map ~:M�215 defined by ~(x, Ez) = (~(x), ~) is C ~. For such an arc, there usually are points (x, ~t) E M � R where % does not satisfy the Kupka-Smale conditions [i3] , [33] along the orbit of ~ through x. In this chapter we shall analyze the behavior of 9 near those points. Such a point is called an elementary bifurcation (point) of 9 (or ~). There are two types of elementary bifurcation points, namely those (x, ~t) for which x is a non-hyperbolic periodic point of %, and those (x, Ez) where x is a non-transversal point of intersection of stable and unstable manifolds of ~ (see [36]). Before we go into details we describe some facts concerning center manifolds in relation to the periodic elementary bifurcation points. Then a description of the types of periodic elementary bifurcations occurring in generic arcs is given. In subsequent sections all these types are analyzed. The saddle-node elementary bifurcation shows some exceptional and unexpected topological properties. In the final section, non-transversal intersections of stable and unstable manifolds are treated. 2. Center m~s,;folds and periodic elementary bifurcations Let {% : M --~ M} be a smooth arc of diffeomorphisms having a periodic bifur- cation at (~, ~). We assume ~to be a fixed point of r if not we replace ~ by 9~ where k is the period of ~-. Let c be the number of eigenvalues of (dq~); of norm I ; since (s ~) is a bifurcation, c ~ i. From the theory ofinvariant manifolds [I2], we conclude the existence ofa "center manifold depending on ~ ", namely a differentiable submanifold W e of M � R such that: -- zw'; O(W ~) n W ~ is open in W ~ (and contains (~, ~)); -- the dimension of W # is c + i and at each point (x, ~) e W ~, W ~ is transversal with respect to M � {~t}; d(~ I W ~ r3 (M X {~}))i has only eigenvalues of norm I. For any k ~ oo we may assume that W ~ is 0 ~ (but as k gets bigger, it may be necessary to take W ~ smaller). However if in some neighborhood of ~ in W~ ----- W ~ n (M � {~)), 13 x4 S. NEWHOUSE, J. PALIS AND F. TAKENS every orbit opt(x) tends to Efor i -+ 4- 0% then W e can be chosen so that W~ is C| even in this last case, W' can not be made (:1% We may consider r I We as a (local) arc of diffeomorphisms; the parameter being the restriction of ~ to W e. Also from the invariant manifold theory [26], [3 o] it follows that r near (E, ~), up to a " local conjugacy ", is completely determined by r I We and the normal data. The normal data consist of the numbers of eigenvalues of (dq~r with norm > i, resp. < i, and of the signs of the determinants of (d~)~, restricted to the maximal invariant expanding, resp. contracting subspaces of T~(M). We say that an arc r is at (~,~) locally conjugate to the arc r215 at (x, ~) if there is a homeomorphism (the local conjugacy) h from a neighborhood of (~, ~) to a neighborhood of (x, ~) such that -- h o 9 ---- ~ o h wherever defined; -- there is a local homeomorphism h a : R ~ R, defined in a neighborhood of ~ such that the ~-coordinate of h(x, ~) equals ha(~) wherever h(x, ~) is defined. So in order to analyze 9 up to local conjugacy at (~, ~) it suffices to analyze 9 [ W e up to local conjugacy; this will be done in the following sections. It should be pointed out that for 9 as above, one can choose invariant manifolds W e, and W ~', the center-stable and the center-unstable manifolds, for ~, containing W e such that the tangent space at (~, ~) is the direct sum of the tangent space of W e at (~, ~) and the maximal invariant subspace of T~(M) on which (dq~)~ has only eigenvalues with norm smaller, resp. bigger, than i. These invariant manifolds are in general not unique. In W e~ and W ~ one can choose invariant continuous foliations o~, and ~-", the strong stable and the strong unstable foliations, such that the leaves are C t, the tangent planes of the leaves depend continuously on the base-point in W% resp. W ~', and such that each leafintersects W e transversally (in W ~', resp. W ~) in one point. Also these foliations are not unique. For details on invariant manifolds and foliations see [i2] and also Chapter IV of this paper. If the leaves of o ~e', or ~-~', have co-dimension one in W~' = W e' r3{tt}, or W~" = W ~ n{~}, then, by [II], the foliation ~"', resp. o~ -"~', is C t. Let 9 and ~ be two arcs of diffeomorphisms with center manifolds W e and W e, let k : W ~ -+ W e be a local conjugacy between @ l We and 9 [ W', and let the normal data of 9 and ~ be equal. Choose invariant manifolds and foliations W e', W '~, o~ TM and o~-~, for ~, and Wee, W ~', ~" and o&" for ~. From [26] it follows that there is an extension H of h to a local conjugacy between 9 and ~ so that H(W e') -~ W e', H(W e~) = W% H(o~") = o~" and H(#-") = ~". In the case of generic arcs r the only periodic bifurcations are those of the following three types (recall we assume ~ fixed): I. c .= i, (dg~)i has art eigenvalue t and the 2-jet of r [ W~ at ~ is different from the 2-jet of the identity; in this case (E, ~.) is called a saddle-node of ~; 14 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS ]5 2. c ---= x, (d~)i has an eigenvalue --x and the 3-jet of q~[ W~ at x is different from the 3-jet of the identity; in this case we call (~, ~) a flip of ~; 3. c = 2, (dg~)i has a pair of non-real complex conjugate eigenvalues on the unit circle and the 3-jet of q~ [ W~ makes ~- an attractor or a repeller, we call these points Hopf points. A periodic bifurcation will be called elementary if it is of type i, 2, or 3 above and it unfolds generically with ~t. This last condition will be explained in sections 3, 4, and 5. Before we consider these cases in more detail, we make some more general remarks: -- in all these cases W~ is C ~~ (because near ~-, each orbit q~(x) tends to 5c as i -+ + oo or as i --* -- oo), but generically it is not possible to choose W c to be C ~~ (we shall not use or prove this fact but see [37]). This results described in this section, except those concerning saddle-nodes are all more or less well-known. Apart form the references in the various sections, one may consult [I], [4], [I9], [2o], [35], [36] 9 3. The saddle-node First we consider arcs of diffeomorphisms { %,} of R (with coordinate x) such that d d ~ d %(0) =o, aTx%(~ = I, ~%(x) 4= o and ~%(o) 4= o (to simplify notation we took ~ = o and ~-= o). Note that if ~ is a generic arc of diffeomorphisms, then ~, restricted to a center manifold of a saddle-node, has the above form. Without loss of d a d generality, we shall assume that ~-~ %(x) > o and ~ %(0) > o. We shall also assume that %(x), as a function ofx and ~, is C 6 and that %(x) is a C ~~ function ofx. Arcs of diffeomorphisms {%} satisfying the above conditions will be called (in this section) saddle-node arcs. We shall prove the following results. Theorem (3. x). -- Any two saddle-node arcs are locally conjugate near (o, o). Moreover the conjugacy can be chosen to be continuously differentiable off the fixed point set. We recall that if{%} is a saddle-node arc, there is a unique C | vector field X, defined on a neighborhood of o in R, such that the time i map X x of X equals % [39]. Theorem (3.2). -- Let (%} and {~,} be two saddle.node arcs with corresponding vector fields X and X, i.e. such that X t = % and X x = ~o. Let h be a local conjugacy between { %} and {~,}. Then h_=hl{x<o,~t=o} and h+=h[{x>o,~t=o} are G ~ and (L).x = x I{ x > o}. Remark (3.3). -- The above theorem implies that the choice of the conjugacy along {~ = o} is extremely restricted. Instead of the usual freedom to fix the conjugacy 15 16 S. NEWHOUSE, J. PALIS AND F. TAKENS arbitrarily on a fundamental domain, we are here only free to fix h in two points: one in {x<o) and one in {x>o}. Let { ~,} be an arc of diffeomorphisms of an n-dimensional manifold M having a saddle-node fixed point at (s ~). Let r t~) = (q~,(x), t~). We say that (Y, ~) unfolds generically if for some (or any) center manifold W ~ at (~-, ~), 9 [ W ~ is a saddle-node arc. Similarly, one can define generic unfolding of periodic saddle-nodes. Theorem (3.4). -- Let ~ be a smooth arc of diffeomorphisms of M, and let (~, ~) be a saddle-node of 9 which unfolds generically. Let W ~ = { (x, -~) [ O~(x, "~) ~ (~, ~) as i -~ oo } (note that W~CW ~, that dim(W') ----dim(W ~) -- I and that W * has a boundary containing (~q ~)). Then the strong stable foliation o~ TM, restricted to W ~, is unique and is preserved under any conjugacy. The proofs of these theorems occupy the rest of this section. As part of the proof of Theorem (3. x), we need to prove the corresponding theorem for vector fields. Consider vector fields X =- X(x, ~.) 0-x on It2 with X(o, o) = o, X(o, o) = o, 0 ~ 0 Ox ~ X(o, o)> o, ~ X(o, o)> o and which are at least C ~. These vector fields are called saddle-node fields. A saddle-node field can of course be considered as a one- parameter family of vector fields on R; its time one map is a saddle-node arc (except for the differentiability); see the beginning of this section. Two saddle-node fields X and X are called locally conjugate if there is a homeomorphism h (compare section 2) from a neighborhood of (o, o) in R ~ to another such neighborhood such that -- h o X t ---- X~o h, whenever defined (X t stands for the time t map of the vector field X) ; -- there is a local homeomorphism h R : It --~ It such that the ~x-coordinate of h(x, ~) equals hs(~. ) whenever defined. First we shall prove: Theorem (3-5). --Any two saddle-node fields are locally conjugate by a conjugacy which is continuously differentiable with respect to x, in the complement of the singular set. Proof of Theorem (3.5). -- Let X and X be two saddle-node fields. For X=X(x,~) 0S we choose a box V ={i x,~) I lxl<a,o<~<_~} such 0 X(x, ~) are positive on U\(o, o). Define f: (o, r ~It+ by that X(x, ~) and ~ XI(n) (- a, 8) = (-k- a, ~), or, equivalently, f(~) ----- f+a (X(x, ~))-Xdx. From this last formula it is clear that lirn0f(8 ) = -4- oo and that the derivativef'(~) is less than zero for 8 ~ (o, r Let U, f, if, ~ be defined analogously for X. Pick o< ~< r so that 16 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS i7 f(0:) 3- max(f(~),f(~)) and take o< ~< ~ so that f(~) =f(~). Then there is a homeomorphism ~q : [o, 0~] -+ [o, ~] such that f(~) =f(~(8)) for o < ~ <_ ~. A conju- gacy h from X to X, restricted to {o < ~ < ~} is now determined by putting h(- = (- a, h(a, = (a, h(o, o) = (o, o); and requiring that hX t = Xth. The continuity of h is automatic except at (o, o). We now prove continuity at (o, o). For any sequence (x~, ~) in U, there are sequences s~, t such that X~/(a, ~q) = xl and Xtl (- a, t~/) ---- xi. (x~,&) ~(o,o) if and only if Ix~--*o, s~--*---o% and t~->+oo. Hence h maps sequences converging to (o, o) to sequences converging to (o, o) which proves continuity. The construction of h [{~t < o} is easy. For example, take h(-- a, tz) = (-- if, ~), h(o, tz) = (o, ~) and h(a, ~) = (if, ix) and extend with hX~ = X~h. On the comple- Oh ment of the set of singularities of X, Ox clearly exists and equals X(h(x, Ix)). (X(x, Ix))-x which is continuous. This completes the proof of the theorem. Let (~} be a saddle-node arc, and let X be the C ~~ vector field defined near o in R such that X 1 ---- ~o. It would be useful if there were a (2 ~ field X extending X to a neighborhood of(o, o) in R' of the form X(x, ~t) = 2K(x, ix) Ox such that X(., Ix)t = %,(.). Although we cannot find such art X, lemma (3.6) provides us with a suitable substitute. We say that a (3 ~ saddle-node field X is adapted to a saddle-node arc {q~a} if the function g(x, ix), defined by (%,(x) + g(x, ~t), IX) = Xx(x , iz), vanishes along the x-axis and has at (o, o) its 4-jet equal to zero. Lemma (3.6). -- For each saddle-node arc {%,} there is an adapted saddle-node field X. Proof. -- q% is C ~~ and has, at x = o, only a finite order of contact with the identity, so by [39], ~0 embeds in a unique C| field ~ = X(x) oX" In general [I4] , [4o] if ~F : (R", o) -~ (R", o) is a C ~~ diffeomorphism with all eigenvalues of (anF)0 equal to i, there is a unique k-jet of a vector field [Y]k, k > i, such that any representative Y ~ [Y]~ satisfies -- the k-jets of Yt and q-' agree at o; -- the eigenvalues of dY at o are all equal to zero. If we apply this to (I) : (x, ~) ~ (%,(x), Ix) at (o, o) we see that the corresponding 4-jet [X] at (o, o) is uniquely determined and has a representative of the form X(x, IX) Ox" 3 I8 S. NEWHOUSE, J, PALIS AND F. TAKENS The restriction of this 4-jet to the x-axis equals the 4-jet of the previously constructed because of uniqueness of that 4-jet (which follows if we apply the above general statement 0 The to q~0 at o). X is now obtained by taking X = (X(x, bt) + X(x) -- X(x, o)) 0x" proof of (3.6) is complete. The next lemma compares high iterates of a saddle-node arc with high iterates of the time one map of an adapted saddle-node field. Lemma (3.7). -- Let{%,} be a saddle-node arc and X = X(x, ~) 0x an adapted saddle- node field which is at least G 5. Let U = { (x, ~t) I o < V. <_ -~, -- a < x < a} be so that for (x, ~) ~ U\(o, o), X(x, ~) > o, ~(x) > x, (~(-- a), ~) ~ U and ]g(x, ~) ] < ~(x) -- x. (Here g(x, Vt) is the function defined just before Lemma (3.6). The fact that the last condition is satisfied for a and Iz small enough follows from [g(x, ~)[ = o(] (x, ~t)[~) and [ 9~(x) -- x) [ > k(~ + x') on { ~t > o} for some constant k.) Then there are constants C1, ~ such that for any (x, ~t) e U, i ~ N, with (~(x), ~t) E U, (i) X~(x, V0 = (9~(x), ~t) for some o~ ~ R with I i - ~ [< ~. C1; (ii) I log((d(9~)x)X(x, ~t)) -- log(X(~(x), ~t))[ < ~.C~. Proof. -- The above two estimates on (high) iterates of ~ are implied by the following two: there are constants K1, K2 such that for any (x, ~t) e U with (~,(x), ~t) ~ U (i)' X~(x, ~t) = (?~(x), V~) for some ~ with I i -- o~ I < p.. (~(x) -- x).Kx; (ii)' [ log((d~)=.X(x, ~)) -- log(X(%,(x), ~t)) I ~ < B. (%,(x) - x).K2. Of course in (ii)' we have to assume that (x, ~) 4: o, but since for ~t = o the whole lemma is trivial we shall assume, in what follows, that always ~t > o. To show that (ii)' really implies (ii) it suffices to observe that d(r (x(x, ~)) = 'rI' d(~)~x)(X(~(x), ~)) x(r ~) ~=0 x(~,~+'(~), ~) " In the following calculation, A will indicate that the formula in which it occurs is valid ifA is replaced by some positive constant, i.e. independent of (x, ~) ~{ (x, tx) [ (x, ~t) ~ U, t~ > o, (q~(x), t~) ~ U}. From the various definitions we obtain IX(x, ~)1 _> a. (x' + ~,) and ]g(x, ~)1 < A.~.(x~ + ~')~ (here one uses that X is G~). Define h(x, ~t) by Xx+~(~)(x, ~t) = (%(x), ~). Since h(x, ix) = f~ IX(y, ~z)]-~dy, where ~ = Xx(x , ~) and [~ = @~,(x), the above inequa- lities imply I h(x, ~)1 < A. ~. (x' + ~'). 18 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS Now (i)' follows from the obvious inequality I ~(x) - x l _> A. (x~ + ~). To prove (ii)', we observe that from previous definitions it follows that (a) the 4-jet of (r -- X at (o, o) is zero (since the 4-jet of X t and r agree at (o, o) ; see Lemma (3.6)) (b) r -- X is zero along {~ = o}. Hence and I xcv~(~), ~)1 ~ ~. ((v.(~))' + ~'), I (a~L. X(x, ~) - x(~Ax), ~)l <_ A. ~. ((~A~)), + ~). so l x(~(~), ~)1 Since (x, tz) ~ (opt(x), ~) is a diffeomorphism, ((?~(x)) ~ + ~) < A(x ~ + B"). From these inequalities it easily follows that I log((do~)~. X(x, ~)) -- log(X(o~(x), ~)) [ <_ A. ~. (x s + ~) <__ A. ~. ($~(x) -- x); this proves (ii)'. Proof of Theorem (3. I). -- The proof of Theorem (3-I) will be obtained by showing that if{%,} is a saddle-node arc and X an adapted vector field (of class C 5) then there is a local conjugacy H(x, tz) = (h~,(x), ~) from {%,} to the time one map X 1 of X. We OH shall also construct H so that Ox exists and is continuous on the complement of the fixed- point set of ~. Indeed, by Theorem (3-5) and Lemma (3.6), this implies (3.1). We take h 0 = H[{~ = o} to be the identity. Let u ={(x,~)lo<_ ~_<~, --a<x<a} be as in Lemma (3.7). Take: I. h~,(--a) =--a for o<__~<~ and hence 2. (h~,(?~,(-- a)), tz) = Xn(-- a, tz) whenever (q~,(-- a), t~) e U; 3" extend the definition of h~(x) to { (x, ~) I o _< ~ _< ~, - ~ < x < ~(- a) } in such a way that it is C x and such that the extension, defined by h~(~(x), ~) = Xdh~(x), ~) is also differentiable in a neighborhood of { (0,,(-- a), t~) [ o <_ rL < g}. Now H [ U is uniquely determined; the continuity of H along { (x, ~) [ x > o, ~ = o } follows from OH Lemma (3-7); also the fact that -- is continuous on U~(o, o) follows from that same lemma. 0x 19 2O S. NEWHOUSE, J. PALIS AND F. TAKENS The extension of H to (tz < o} is simple: see the proof of Theorem (3.5). Proof of Theorem (3.2). -- Let { %,} be a saddle-node arc and X an adapted saddle- node field, of class C 5, defined in a box U as in Lemma (3.7). We take -- a < x < o and o < ~<7 a and show how {%,) gives rise to a canonical homeomorphism T~ from a neighborhood ofx (in R) to a neighborhood of~. These maps T~, are called translations. Let {xi}i~ ! be any sequence converging to x; choose a sequence {t~}~! such that ~(xi) converges to ~- with ~ -* o as i --> oo. To get such a sequence, choose ~t~ -+ o as i -+ oo such that (X~.) (x o ~q) -+ (~-, o) and apply Lemma (3.7). Let x' be a point close to x; without loss of generality we may put (x', o) -- (X~) (x, o) for some ~ e R. Then for any sequence x~ -+x', lim i ' 9 %i(x~) (the same ~q's as above) exists and equals |-+ a0 (X~)(~-, o). This can be seen as foUows: for some ~, ~, ~ we have (~(,,), ~,) = x~,(,,,, ~,); i *' t (~(~,), v,) = x~(x,, ~); x~,(x,, ~,) = (x~, ~). By Lemma (3.7) I i -- ~,1 < ~,.Cx and Ji -- ~il < re.C1 for some constant C1; because x i ~x, x; ---)-x' and X~(x, o) = (x', o), 0t i-+a. From this we conclude that if ~ (~/(xi) ~ ' , then = + ~ -- ~ so lim ~i= a. We define x~,(~,(x~), w) = v~) ~', :~, i--r oo T;.(x') lim i ' = ~(x~). It is clear that the definition of the local homeomorphism T~ is independent of X: X was only used to show that lim ~ ' i-.~o ~/(xl) exists. Observe that (Tz.).(X) = X. Now we extend the notion of translation. If x, ~- are on the same side of o and x' on the other side, we may define T;~ = T~=, o T~,=; this definition is independent (at least near x) of the choice of x'. Hence we see that the translations form the pseudo-group of all local diffeomorphisrns of (-- a, + a)\{o} which preserve Xl{~=o}. If h is a local conjugacy between two saddle-node arcs { %,} and { ~,,}, it must also conjugate the translations defined by {%} with those of{~,}. Let X, X be the smooth vector fields such that X x=q~0 and X t=~0. For x#o, ~-4=o and z=X~(x) near x, let x' = h(x), ~' = h(~), z' = h(z) and z' = Xv(x' ). Since h conjugates T~, and Ti,,,, and T~(z) = Xt(s Ti,~,(z') = ff, t,(~'), we conclude that there is a (conti- nuous) function t'= a(t) such that hXt h-t= "Xol0" From the group property Xt, o X~, = Xt,+t, we deduce that or(t) is linear in t. Since Xx = ~0, Xx = ~0, we have h o X t o h- t = Xx and hence h o X~ o h- t = X~ for all t. This proves the theorem. Proof of Theorem (3-4). -- From Theorem (3. I) and invariant manifold theory (see Section 2 and [26]), it follows that ~, near (s ~), is locally conjugate with ~ near (o, o) where g(x~,...,x,,~)= x,+~+~,247177 . 20 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 2x The stable manifold W* of 70 is {(xt, ..., x=) : x x < o}. We take the strong stable foliation #" of ~ near (o, o) to consist of the manifolds { (xt, ..., xm, ~) : (xx, ..., xh, [z) = constant}. In a neighborhood U ofo, we give a dynamical characterization of the leaves of ,~** c~ ~': qx, q2 e W' belong to the same leaf of ,~" if and only if there are sequences { [zl}ieN, { ~)~e ~ and { q~}~eN such that -- (6, ~) -+ (qt, o) and (6, ~) "-* (q,, o), -- for each i, {~'~(ql) ' and }~-0 ' are contained in U; 9 ~'~ i 9 ~i " -- lim(~(qt), ~) = lim(~m(~), ~t~) artd this limit is not the saddle-node point. This follows from the special form of ~. Let ~'~' be some strong stable foliation for ~. The local conjugacy between and ~ mentioned before cart be chosen to map o ~'~' to o q~''. This implies that o~"' ~ W' also satisfies the above dynamical characterization. Hence, ~"' t3 W' is unique and is preserved under conjugacies. Theorem (3.4) is proved. Note that a local conjugacy of~0, or ~, does not have to respect the above foliation. This has to do with the fact that the dynamical characterization was only possible by using 9 on a full neighborhood of (o, o) in R '~ � R. 4. The flip Here we consider arcs {q~) of diffeomorphisms on R such that ~0(o) = o, (dq~0)0 = -- I and such that the 3-jet of (~0) ~ at the origin differs from the 3-jet of the identity. Such arcs of diffeomorphisms are obtained by restricting a generic arc of diffeomorphisms, at a flip, to a center manifold. The origin is either a source or a sink of q~0; in the following we shall assume it to be a sink of ~0; the other case then occurs for ~o t and is completely analogous. With a coordinate change of the form = ~'(x, ~) we can put r in the form = - + oct') + oCl l. ') where X is a real function and X(o) = o. We shall say that the flip unfolds genzrically dX if ~-~ (o) 4= o. This is, of course, a generic condition. In higher dimensions, we will dX use this teriidnology if ~-~ (o) 4= o on some center manifold. Similar considerations apply to periodic flips. dX Now we return to the one dimensional case. We shall assume ~-~ (o) > o; the 21 o~ S. NEWHOUSE, J. PALIS AND F. TAKENS other case can be reduced to this by replacing "~ by --~. It is clear that the phase portrait looks as follows: ..,.Icf Fixed points \ ,1/ t Points with period two >t J 17161. I The foUowing theorem can be easily proved using the methods of proof used in Theorem (3.5) in the region {~ <__ o}: Theorem (4-x). -- Any arc {%} of diffeomorphisms on It which is of the form %(x) = -- x + x 3 + ~.(p.).x + o(x 4) + o(1 ~[ .x ~) with Y(o) > o is locally conjugate with = - x + x + v.x. 5. The Hopf point We consider arcs {%} of diffeomorphisms of R ~ such that %(0) = o, (d~0)0 has eigenvalues on the unit circle, but different from + z, and such that the 3-jet of 9o makes the origin an attractor or a repeller. We shall assume it to be an attractor; otherwise we consider 9~-1- Up to a change of coordinates, the origin will be a fixed point of % if I ~[ is small. Let ),(~), X(~) be the eigenvalues of (d%)0. For generic arcs one has [),(~)l~-0 :l: o, and we shall say, in this case, that the Hopf orbit unfolds generically. As for the saddle node and flip, we shall use the same terminology for periodic Hopf points if they unfold generically on center manifolds. In the following, we return to dimension two, and we shall assume that ~ Ix(w)[ _o > o; otherwise we replace by -- ~t. These arcs have been extensively studied; for references see [32]. From the fact that, for tt <__ o, the origin is an attractor of % one concludes: Proposition (5. x ). -- If{%} and { ~,} are arcs of diffeomorphisms of R', satisfying all the above requirements, then there is a continuous one parameter family of homeomorphisms h~, ~ < o, from a neighborhood of o E R" to a neighborhood of o ~ R 2, such that h~, o % = ~, o h~, when- ever defined; i.e. (x, ~) ~ (h~(x), ~) is a local conjugacy between % and ~, on ~ <__ o. 22 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 2 3 A consequence of Proposition (5- I) is that arcs {%} as above are locally stable for ~<o. For ~>o this is not the case. Theorem (5.~). -- Let {%,} be a Coo arc of diffeomorphisms as above. Then in any Coo neighborhood of{%} there is an arc {~} such that the arcs {%}, {~} are not conjugate, even not mildly conjugate. Proof. ~ We can first approximate the arc {%} by an arc {~} such that the eigen- values of (d%) 0 have the form e 2"~ with 0r irrational Then we know [4o] that ~',, is rotationally symmetric in the formal sense. By this we mean that there is a smooth S 1 (= R/2~) action R on R~� R, differentiably conjugate with the usual action R : R,(xl, x2, a) = (xl.cos 0r + x~.sin ~, -- x 1 sin ~ + x~ cos ~, a), such that for each ~, R, o ~ and ~ o P,~ have the same oo-jet in the origin. So with a second perturbation one can find an arc { ~'~,} such that the corresponding map ~' defined by ~'(x, ~) = ~(x) commutes with the S x action R, at least on a small neighborhood of (x ---- o, ~ = o). One knows [32] that ~, for a > o, has an invariant circle, say C,. Because ~' commutes with the smooth S 1 action (for a small), ~ [ C~, is differentiably conjugate to a rotation. Since there is a residual subset of the diffeomorphisms of S x no element of which is conjugate to a rotation, there is finally an approximation ~ of ~ (in the C oo sense) such that ~ also has C~, as invariant circle but such that for some sequence { al}~e i, a~ -+ o as i ---> 0% ~ [ C~, is not conjugate to a rotation. It follows from the construction that { ~,} and { ~} are not conjugate and even not mildly conjugate near (o, o). Hence % cannot be (mildly) conjugate to both {r and { ~}. This proves the theorem. Remark (5-3)- -- If{ %} is an arc of diffeomorphisms on a manifold M of dimension m > 2 which has a Hopf point at (~, ~) then the conclusion of Theorem (5.2) remains valid. In the construction of the C ~~ arcs { ~-~}, {qS~}, one has to add in this case a preli- minary step, namely one has to modify { %} first so that it has a Coo centermanifold. Then the modifications, as described in the proof of (5.2) are carried out in that center manifold and extended to a neighborhood. 6. Quasi-transversal intersections We consider a C ~~ arc of diffeomorphisms {%: M-+M} and assume that for some ~ e R and some ~ e M, there is a non-transversal intersection at ~- of a stable and an unstable manifold (of some periodic points) of q~g. In such a situation, if we denote the stable, resp. unstable, manifold by W', W ~, there is a canonical quadratic map D : T~(W') c~ T~(W") -+ T~(M)/T;(W') + T~(W"), which is analogous to the intrinsic 2 nd order derivative [3] and is defined as follows. Let i: T~(W ~) t3 T~(W") -+W ~ be a smooth map such that i(o) = ~ and (di)o is the canonical injection; let z~ be a projection (i.e. ~ = ~:) of a neighborhood U of ~- 23 S. NEWHOUSE, J. PALIS AND F. TAKENS to a submanifold whose dimension equals the dimension of T~(M)/(T~(W') + T~(W")), and such that 7r(~) = ~-, (dTr)~(T~(W 8) + T~(W")) = o and 7r(W*) = k-; let N = re(U). Then T~(N) is canonically isomorphic with T~(M)/(T~(W') + T~(W")). The map 7r o i maps o to ~-, d(~ o i)0 = o; hence the second derivative dg(~ o i)0 : T~(W') c~ T~(W") -+ Ti(N) is a weU defined quadratic map; we define D to be the composition of d~(Tr o i)0 with the canonical isomorphism T~(N) ~ T~(M)/(T~(W') + T~(W")). The maps D does not depend on the various choices. It is not hard to show that, for generic arcs of diffeomorphisms {q~}, all the non- transversal intersection points ~ofstable and unstable manifolds of ~, for ~ E R, s ~ M, will satisfy (in the above terminology): a) dim T~(M)/(T~(W 8) + Ti(W")) = I ; b) D is non-degenerate. Under these circumstances there are coordinates xx, ..., x,, on a neighborhood of x- (~ corresponding to o) such that W' and W ~ locally have the following Canonical form (6..) W 8 = (xx ..... x~_, = o} W" = (x.+~ ..... x,. = o, xl =f(x,._,+...., x.+d} where m = dim(M); s = dim(W'); u ~- dim(W") ; f is a homogeneous non-degenerate quadratic function; if m--s+ x>u+ I then one should read x 1=o. Before proving (6. i), we note the following c) max(s, u) < m by condition (a) above; el) the vector is not in T~(W') + T?(W"); 0 0 , .. are in T~(W') c~ T~(W"). e) the vectors ~x,,_-,+l ~ "' ~x~-+l we can choose a Proof of (6. i). -- Since dim(T~(W ~ + T~(W")) = m -- r, From this it clearly (u + 1)-manifold W" containing W" which is at ~- transversal to W'. follows that one can choose coordinates ~-1, ..., ~,, such that w' = (~1 ..... ~_, = o}; ~" = (~.+2 ..... ~. = o}; W" is tangent at ~- to {xx = x-.+~ = x-.+ s ..... ~-,. = o}. 24 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS ~5 Then, for some functionf w" = {x-,+~ ..... ~ = o, ~ =f(~,,..., ~+d}, f(o)=o and (df)o=o. Now we replace ~t by the new coordinate Xl = "~1 --f(x-9., 9 ..1~ g'-u+l) +f(o, ..., O, X-m--,+l, "'', X-u+l); then we have W m ~ {x 1 ~ ~9. ..... X-m--s ~ O}~ W ~ = {xl =f(o, ..., o, ~-m-,+ 1, -.., x~+l), ~+~ ..... ~m = o}, the first derivative of (x-,~-~+l, ..., x,+l) ~f(o, ..., o, ~,~-,+1, ..-, x-~+l) is zero and the second derivative is just D and hence has maximal rank. Now we can apply Morse's Lemma [3] and obtain a coordinate change of the form xj=~ for 2<j<m--s or u+2<j<m, x~ -= xj(~m_,+ l, . . ., -~+ x) for m -- s + i < j <_ u + I, for which we have W' = {x~ ..... x,,_, = o}, W" = {x~ =f(x,,_,+l, ..., x,+l), x,+, ..... x~ = o), with f homogeneous, quadratic and non-degenerate. Remark (6.2). -- In suitable coordinates the generic unfolding of the quasi- transversal intersection puts the manifolds W~, and W~, in the form W~ = {xl = o ..... x~_, = o), w~ = (x,+, ..... x,, = o, x~ =f(x,,_,+~, ..., x,+d + (~ - ~)). 4 III. -- NECESSARY CONDITIONS FOR STABILITY OF ARCS In this chapter we shall obtain necessary conditions for the various kinds of stabilities of arcs we have defined. Let I = [% x] and let t~ be the space of C | arcs q~ : I -+ Diff~~ with the G ~176 topology. Elements of~ will be denoted by ?, or by { %) when we wish to make the dependence on the parameter t~ ~ I explicit. Recall that ,~ C ~ is the subset of those arcs ? ~ ~ such that (I) '~o E MS, (2) b = b(~) = inf{~ EI : ~. CMS}< i, (3) the limit set of ?~ consists of finitely many orbits. A diffeomorphism f is called elementary if either (a) there is exactly one quasi-hyperbolic periodic orbit, the other periodic orbits are hyperbolic, and all stable, strong stable, unstable, and strong unstable manifolds meet transversally; or (b) all periodic orbits are hyperbolic, there is one quasi-transversal orbit of intersections of stable and unstable manifolds, and all other stable and unstable manifold inter- sections are transverse. Let ~ C ~ be the set of arcs ~p ~ M such that % is elementary and the quasi- hyperbolic periodic orbit of ~, if it exists, unfolds generically. We proceed to discuss necessary conditions for stability of arcs in ~. x. The modulus condition (quasi-transversal intersection) In this section we shall show that left stability of an arc { %} in ~ forces all stable and unstable manifolds of periodic points of ~b to meet transversaUy. Before this, we consider the effect of a quasi-transversal orbit on topological conjugacy. The next theorem shows that generally such an orbit yields at least a one-parameter family of distinct topological types. We will refer to this phenomenon by saying that moduli occur. It should be noted that this occurs even in a locally isolated codimension one submanifold of the boundary of MS on the 2-sphere (hence, on any manifold of dimension larger than one). 2fi BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS For example, consider an MS diffcomorphism % on S z as in the next figure. 2, /\ ~> /\ 0 < 0 > < \/P2 ];'10. 2 The circles represent sources and sinks and there are two saddle fixed points Px and ps. We choose a curve of C s diffeomorphisms {%}, o <~ tL _~ t, starting at % so that px andpl remain fixed for each %, and W"(pt, ?~t) has a single orbit r of quasi- transversal intcrsectiom with W~ 9~) as in the next figure. I/2 (X) o~) 2 fx) ~. (~ ~ / P2 / \ / Fro. 3 This can be done so that % is in MS for ~ 4= - and any perturbation {q~} of{%} has a unique bifurcation b(9' ) near -. Let f be a diffeomorphism of M with a hyperbolic fixed point p. Let =" be the largest modulus of the eigenvalues of dr(p) which are inside the unit circle, and let 8" be the smallest modulus of the eigenvalues of dr(p) which are outside the unit circle. If there is an eigenvalue = of dr(p) such that (x) lml = ~', (2) ~ has multiplicity one, (3) any eigenvalue X ofdf(~) different from ~ or the complex conjugate i satisfies [ ), [ 4= ~*, then we say that the weakest contracting eigenvalue = off at p is defined. Similarly, 27 28 S. NEWHOUSE, J. PALIS AND F. TAKENS if there is an eigenvalue [~ of multiplicity one of dr(p) such that ] ~ [ = [~* and any eigen- value X 4: ~, [~ satisfies [ X 1 4: [~*, we say the weakest expanding eigenvalue [~ off at p is defined. Let p and q be hyperbolic fixed points off such that the weakest contracting eigen- value ~ at p and the weakest expanding eigenvalue ~ at q are defined. Let t-I v be a C 1 invariant manifold containing W"(p) and tangent at p to the sum of the eigenspace of ~ and TvW"(p), and let I-Iq be a C t invariant manifold containing W'(q) and tangent at q to the sum of the eigenspace of ~ and TqW'(q). The existence of I-I v and Hq is.proved in [I2]. They are not unique. However, the tangent bundle of Hp along W"(p) is unique, and so is that of Hq along W'(p). Let r be a quasi-transversal intersection of W"(p) and W'(q). We say that r is a regular quasi-transversal intersection if W"(p) is transverse to Hq at r and W~ is transverse to Hp at r. This definition is independent of the choice of the manifolds H~ and Q.q because it depends only on THp, THq along W"(p), W~ respectively. Note that part of the definition of a regular quasi-transversal intersection of W"(p) and WS(q) is that the weakest contracting and expanding eigenvalues ~ at p and [~ at q be defined. Note that for arcs q~ in a residual subset of ~ all quasi-transversal orbits of %, are regular. Theorem (t. i ). -- Let f, f' be C 2 diffeomorphisms. Let p and q be hyperbolic fixed points having an orbit {f~(r)} of regular quasi-transversal intersections of W"(p) and W'(q). Let o~ be the weakest contracting eigenvalue off at p and ~ be the weakest expanding eigenvalue of fat q. Make analogous assumptions on f' concerning fixed points p', q', eigenvalues ~', ~', etc. If there is a conjugacy from f to f' defined on a neighborhood of the closure 0f{fi(r) } mapping p to p', q to q', and r to r', then log 10: I log 1~'1 log log I '1" Remark x. -- Ifp and q are periodic points of period x(p) and ~(q) instead of fixed points, and n is the least common multiple of v(p) and x(q), the theorem can be applied to f". Doing this one obtains that if ~ is the weakest contracting eigenvalue off "Ip/at p and [~ is the weakest expanding eigenvalue off "cq~ at q and 0(, ~' are the corresponding eigenvalues for f', then the existence of a conjugacy between f and f' implies log I~ I log I~'l logl l Iogl 'l 2. Theorem (I. I) shows that quasi-transversal orbits lead to at least one dimen- sional invariants of topological cortjugacy. It is interesting to ask what additional invariants in the presence of such orbits are sufficient to imply the existence of a topo- logical conjugacy in various contexts. For some results in this direction, see [I8], [28]. 28 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS Proof of Theorem ( x . i ). A) The case m=2; re=dim(M) In this case the dimensions of the stable and unstable manifolds are both one. Replacing f, f' by f~, f,2, if necessary, we assume ~, 6, ~', 6' > o. So we are in the following situation: __ W u (p~ W s ~ql \/ /\ FIG. 4 We consider a sequence of points rl converging to r but so that r~ r (W"(p) u WS(q)) for all i. By choosing a subsequence if necessary, we can arrange that there are sequences of integers n~ ~ oo, ml -+ oo with the property that f-'~(r~), f'~(ri) , has a limit in WS(p) -- p, W"(q) -- q, respectively. Let p(ri, W"(p)), p(r~, WS(q)) denote the distance from r i to W"(p), respectively W"(q), with respect to some Riemannian metric. Because f is C 9", it is C 1 linearizable on WS(p) and W"(q) [8]. From this, we conclude that p(rl, W"(p)) ~ e'~ and p(ri, W"(q)) ~ 6 -~, where ~ denotes equality up to a positive multiplicative factor, depending on i but uniformly bounded and bounded away from zero. It is clear from the picture, or rather from the normal forms in Chapter II, that a sequence ~ can be chosen so that p(rl, W"(p)) ~ p(r~, W"(q)). In that case we have log 0~ _ lira mi log 6 i-~| n~" Now we assume that there is a local conjugacy h, defined on a neighborhood of the closure of the orbit of r, as in the theorem. Let h(r~) ----- r'~. From the topology of the intersection of W ~ and W * and the position of the r~'s (see figure) it follows that p(r~, W"(p')) < p(r;, W'(q')); i.e. p(r , w"(F)) < i. p(r , w'(r - 29 3o S. NEWHOUSE, J. PALIS AND F. TAKENS Since (f')-'~(r~) and (f')"~(r~) must have a limit in W'(p') --{p'} resp. W"(r --{r wc conclude that p(r~, W*(p')) ,-~ (Qt')~ and p(r~, W~ ,~ (f~')-=,. This, together with p(r , < i pCr , w'(()) - implies that log oc' m, log ~ log oc' > log <~ -- lim -- = --. Reversing the argument, one finds log ~' -- log log ~' -- ~-,~o n~ log and hence log ~' log 0~ log ~' = log ~" Observe that if W"(p) is transverse to W~162 and W"(p') is transverse to W'(q'), then we cannot conclude for any sequence r~ -+ r that I~(r~, W~(P')) is bounded. So p(r , w'(q')) we needed some of the properties of the " topology of the intersection ". In the next higher dimensional case we have to analyze this in detail (1). B) The case dim(W"(p)) = dim(W"(r = m -- x As in case A we assume a, ~, 0~', ~' > o and we consider sequences ri --* r such thatf-~(rl) converges to a point in W'(p) --{p}, and f~(ri) converges to a point in W"(q) -- {q} for some sequences nl, m~ -~ co. We also assume that one of the following three possibilities takes place (this can be obtained by taking a subsequence): p(r. W"(p)) is bounded and bounded away from zero (or ,~ x); p(r,, w'(q)) --, co. In these cases we find that < lim inf(~) i.-* oo -- ; log fi ~-'| -- ; > lira sup -- . -- i--*. eo We assume that there is a local conjugacy h and denote h(r~) by r~. We assume also that p(r~, W"(p')) either goes to zero, is ~-, I, or goes to co; in each of these cases we get w'(q')) log ' tim" a relation between ~ and " -- as above. (~) After the arguments for sections B, C, and D were written, a cleaner treatment was discovered by S. VAs STRreN. This treatment uses arguments pre~ented in the proof of Theorem (2.1) in lisa]. 30 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS log ~ < log ~' From this we claim that, if ~ log ~" then, for each sequence r~ -+ r as above, pCr,. w"(l,)) p(r., w"(l,')) ~- o and/or ~. oo. 0(r,, w'(q)) p(~;, w'(q')) log ~ log ~' Indeed suppose that ~ < log ~' and r~ -~ r as above. If does not go to 0% then for some constant 0 > o, we have (~')'~(~')'~ <_ c. This gives n~ log ~' + m~ log ~' <__ log C or log ~' log G m~ log ~' -- n~ log ~' hi" If 8> z is so that log0c+log8 log~' = -- then log ~ log ~" log < log C rn~ log ~ -- n~ log ~ n~' or n i log ~8 + m~ log [~ < log C log -- " log [$'" This means that (a~)~, is bounded, so ~'~ ~"~-+ o as i --~ oo which implies that p(r,, w"(p)) 9 ---~ O, p(r,, w'(q)) logoc log:C This proves the claim. Similarly, for ~ > log ~' 9 ,~e have p(~,, w-(p)) p(,;, w-(p')) oo and/or --> o. p(r,, w'(q)) p(~, w'(q')) We want to prove that this cannot happen for every sequence in case of a quasi-transversal intersection of W"(p) and W'(q) (and W"(p') and W'(q')). For this we take subsets M, A ~, A ~ of a neighborhood of r such that: -- A' u A " u A ~ is a neighborhood of r; ~(,,, w"(p)) -- if rl-',-r is a sequence such that -+o, then r~A ~ for i big; p(r,, W'(q)) if r i ~ r is a sequence such that p(r~, W"(p)) ~.o% then r~A ~ foribig; p(r,, W'(q)) r~ r WU(p) t3 W'(q)) then --if r i-)-r isa sequence such that r~eA ~ (and p(n, w-(p)) e'~a I ~ p(r,, w'(q)) a_____~ W'(q'))W*(P')) p(r~,P(r~' s. NEWHOUSE, J. PALIS AND F. TAKENS A TM, A" and A '~ are similar sets in a neighborhood of r'. From the above arguments _ log ~' it follows that if log 0t < then there is a neighborhood U of r such that log ~ log [3" h(U n (h ~ u A')) C A" log ,t log a' (and if ~>log~,, h(U n (A ~ w A")) C A'"). In order to derive a contradiction from this (namely from the assumption that log a< log 0d I in the case of quasi-transversal intersection, we proceed to an explicit log ~ log ~'] construction of A ~, A' and A ~ From Chapter II we know that there are coordinates xl, ..., x,~ in a neighborhood of r such that locally W'(q) = {x. = o} and W"(p) = {x m = ~(xl, ..., xm_x)} where ~ is a homogeneous quadratic function. In these coordinates we take { , } A': Ixml <_ g I ~(xx, 9 9 ~m_x)l A" = Ixm -- ~(Xl, ..., x,~_l)l < g I ~(xl, ..., xm_l)l / ' I A ~ == closure of the complement of A' u A ~. Near r', W~(p ') and W"(q') have the same form, so there is a diffeomorphism ~" from a neighborhood of r to a neighborhood of r', mapping W"(p) to W"(p') and W'(q) to W'(q'). We define A" = LF(A'), A '~ = ~(A ~) and A '~ = ~(A~ It now follows that ~ = ~- 1 o h is a local homeomorphism from a neighborhood ofr toitself, inducing homeomorphisms in W"(p) and W*(q) and mapping (A ~ u A*) n U into A'. Let U*CU be a subset of the form U* ={(xt, ...,x=) [o< Y.~< a*} and U**D~(U*) a set of the form U** ={(xz, ...,x,,) l o< Y~x~< a**}. From the fact that the subsets W~(p), W*(q), A ~, A' and A ~ restricted to U* or U** are all cone- formed, and the fact that ~ is a local homeomorphism inducing local homeomorphisms in W~(p) and W*(q) we conclude that the maps o" n W'(q) -~ u'" n w'(q), o'- (w"(p) n u') -+ o"- (w"(p) n u"), induced by ~, induce isomorphisms ill the homology. From the definitions it is clear that the inclusions U' n W'(q) -+ U* n A', U" n W'(q) ~ U** n A', U" n (A' w A ~ ~ U* -- (W"(q) n U*), U** n (A' w A ~ ~ U** -- (W"(q) n U**), 32 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS are all homology equivalences. Now consider the morphism of long exact sequences: H,(U* n M) ~ Hi(U* c~ (A* u A~ ---> Hi(U* n (A' o A~ U" n A') -~ H,(U*" c~ A') -+ H,(U*" n (M o A~ -+ H,(U" n (A' w A~ U*" n A) hi,/h and ha axe all induced by ~, so hx,/h are isomorphisms. By the Five Lemlna, ha is then also an isomorphism. But since ~(U* c~ (AS u A~ C U** c~ A", ha is the zero morphism. Hence HI(U* c~ (A* u A~ U* c~ A *) = o for all i. This is in contra- diction with the following result, which follows from standard arguments: if ~(xa, ..., x,,_~) = ~ -t- ... + ~-- 4+~--... -- xs,,-t then Z if i=k or i=m--k--I provided k4:m--k--i; H~(U*n(A'uA~ ZSZ if i=k=m--k--I; o otherwise. C) The general case: ~, ~, od, ~' real We assume that f has fixed points p, q and invaxiant manifolds Hp and Hq as in the theorem and ~, [~, 0:', ~'> o. Then, dimHp dimW"(p) + i, and dim Hq = dim W"(q) + i. Let Hp,, Hq, denote the analogous manifolds for f'. Of course, if there is a conjugacy h betweenf andf', it does not follow that h(Hp) = Hp, or h(Hq) = He. If we knew that h(Hp) ---- Hv, and h(Hq) = Hq,, we could apply the argument of case B for sequences r i->r in Hpr~Hq. For in HpnHq, the manifolds WU(p) n Hp (~ Hq and W"(q) n Hp n Hq have codimension one, and intersect " nearly " quasi-transversely. We say nearly because, since Hp and Hq are only C t manifolds, a quasi-transvelse intersection of WU(p) n Hp ~ Hq and W'(q) n Hp n Hq is not defined. However, we Call choose C 2 submanifolds Hp and Hq (not necessarily invaxiant) which are G I close to Hp and Hq, respectively, such that Hp n ~p D WU(p), Hq c~ Hq D W*(q), W"(p) n H, n Hq = W"(p) n H, n Hq, W"(q) n Hp n Hq = W"(q) r~ Hp n Hq, and W*(q) c~ Hp n ~q intersects W"(p) n Hp n Hq quasi-transversely. Hence, the metric properties of the intersection of WU(p) c~ Hp r~ Hq and W*(q) c~ Hp r~ Hq in Hp c~ Hq are those of a quasi-transversal intersection. Since h does not map Hp to Hp, or Hq to He, we must modify our arguments. We shall show that there is a map  from h(Hp n Hq) to H, n He such that for any 5 S. NEWHOUSE, J. PALLS AND F. TAKENS x 9 h(Hp n Hq) -- (W"(p') u W'(q')), we have p(x, W"(p')) ,-~ p(~(x), W"(p')) and p(x, w'(r w'(r Let N be a small disk neighborhood ofp where we can define a C 2 coordinate chart such that the components of WS(p) n N and W"(p) n N containing p are coordinate planes. In the sequel, we will xestrict ourselves to such a neighborhood N and to the components of W"(p) and W~ containing p. Let % : N -+ W'(/~) be the natural projection and assume that the negative orbit of r lies in N. Since W'(q) is transverse to Hp at r, we can characterize W~ as the subset of W~ where the critical points of ~, ] W'(q) accumulate. This characterization is based in the following fact: If v 9 W~ then v r W"(p) if and only if there are a neighborhood V of v and a neighborhood U of r such that for each v' 9 V n W~ there is an arbitrarily small neighborhood V' of v' in V such that, for n 9 N suficiently big, {f-"(W~ n U) n V'} C{f-"(W~ n U) n V} is a homotopy equivalence. In this formula we should omit from f-"(W'(q) n U) those points whose orbits from W~ n U to f-"(W~ leave N. From this dynamical characterization it follows that h(W"(p))= W~ ') and similarly h(w""(q)) = w.-(q'). Thus h(Hp) n W'(p') is an invariant C o curve off' which meets W"(p') only at p'. Sincef' is C t linearizable on W~ ') [8], we see that h(Hp) n W~ ') has to lie in a small sector about Hp, n W'(p ') which has width zero at Hp, n W"(p') as in the following figure. >> << W `s (p') J L Hp, n W s (p') Fm. 5 34 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS From this it follows that for any smooth projection ofa neighbourhood of Hf to Hv, , the distance of points in h(Hp,) to W"(p ') is not changed essentially, i.e. the quotient of the new and old distances is uniformly bounded away flora zero and infinity. It is not hard to choose such a projection al on Hf which does not essentially change (at least near r') the distances to W'(q'). Using the same type of arguments one can define a smooth local projection a, on H e, in such a way that -- ~ maps Hp, into itself -- ~, does not essentially change distances from W~(p'). As before, it follows that ~2 does not essentially change the distances from points in h(Hq) to W~(q'). So ~2o~1[ h(HpnHe) :h(HpnHe) -+Hp, c~H e, has the required properties. To see that the whole reasoning of case B applies, we first choose metrics on Hp, H, H ,, H e, induced by nearby C 2 manifolds Hp, Hq, Hf, He', respectively. Deft- rung A', A ~ A" C Hp c~ He as before and projecting them into Hp and H e to obtain A 8, A ~ A", we then have that the maps W"(p) n Hp n H e -+ A ~, W'(q) c~ Hp c~ Hq -+ A', etc., are homotopy equivalences. Now the argument can be completed by observing that (I) (~ roughly preserves distances; (2) a induces, locally at r', homotopy equivalences from W'(q') n h(Hp n I-Iq) to W'(q') c~ (Hp, n He, ) and from h(Hp n Hq) -- W"(p') to (Hf n He, ) -- W"(p'); (3) if r~-+ r', r~ e h(Hp n He) are such that f-'~'(r'), respectively aOm(r~), has a limit in WS(p ') --p', respectively WU(q ') --q', then o(r[,W~(p')) ,~, (0d)m and v(r;, W'(q')) These facts follow easily from the constructions. D) The general case: ~, od are not real or ~, ~' are not real Let Q be the df(p)-invariant subspace of TpW'(p) complementary to the eigenspace of 0c, and let Q be the df(q)-invariant subspace of TqW~(q) complementary to the eigenspace of ~. Let W'8(p) be the invariant manifold in Ws(p) tangent at p to Q., and let W*"(q) be the invariant manifold in W"(q) tangent at q to Q. If Hp and Hq are the invariant manifolds in the statement of Theorem (I. I), then W**(p) is transverse to Hp at p and W~"(q) is transverse to Hq at q. The fact that W"(p) is transverse to H e at r enables us to get a dynamical characterization of W""(q). Similarly, we will obtain a dynamical characterization of W"(p). Once these are obtained, it will follow that a conjugacy h from f to f' as in the statement of Theorem (i.i) will have the property that h(W'(p)) = Ws'(p ') and h(W""(q)) = W""(q'). From this, it will follow as in case C that h(H e n W'(p)) is in a small sector about Hp, t~ W"(p') which meets W"(p') only at p'. Similarly, h(H e c~ W"(q)) is in a small sector about H e, n W"(q ') which meets W'"(q ') only at q'. The arguments then proceed as in case C. In the present 35 S. NEWHOUSE, J. PALIS AND F. TAKENS situation either or both of W~(p) and W'(q) will have codimension 2 in Hp and Hq, respectively. This changes the arguments slightly, but, since these changes are straight- forward, they will be left to the reader. Thus to complete the proof of Theorem (x. I), we must show how W~"(q) is characterized dynamically. Let U be a small disk neighborhood of q containing the forward orbit of r. We can take U as a coordinate chart with a Cl ~ coordinate system having the components of W~(q) and W*(q) containing q as coordinate planes. Let ~ : U -+ W~(q) be the natural projection. For an integer n> o let ft~ denote fn] N f-J(U). Define 0<j<n W~(q) to be the set of points y ~ W"(q) n U such that (.) for each sequence o < nx< na< ..., the critical points of ~, ]f~(W"(p)) accumulate on y as i ~ oo; in case dim(W"(p)) + dim(WS(q)) = dim(M) -- I), f~(W"(p)) accumulates on y. We assert that W~"(q) =U ~>0f'( w" to~(q)). This implies a dynamical characte- rization of W"(q). To prove this assertion, first suppose that flU is linear and, near r, the set of critical points (or fold points) L of ~, ] W"(p) is an affine subspace. Here we think of U as an open subset of Euclidean space via linearizing coordinates. Note that dim(L) = dim(W"(q)) -- i. If the eigenvalue ~3 is real, the assertion follows as in the previous case when the eigenvalues 0~, [~ were taken to be real. So assume ~ is not real. Assume also that Hq, W"(q) and W"(q) are linear subspaces of U near q. Choose an affine subspace L 1 C L which is complementary to Hq at r. Then f"(L1) converges to as n -+ or. This implies that (.) holds for any y e W~(q) near q. Now let y be in W"(q) -- W~'(q) and near q. Since f] Hq n W"(q) near q is a rotation, there is a sequence of integers n 1 < n~ < ... such that f~i(L n Hq) does not accumulate on ~y as z -+ oo where 7~ : W"(q) ~ W~(q) n Hq is the natural projection. This implies thatJ~(L) does not accumulate ony as v -+ or. Tiffs proves the assertion if f] U is linear and L is affine. The extension to the case wherefis not linear or L is not affine near r is straightforward. We leave the details to the reader. Theorem (i. I) is proved. Lemma (1.2). -- Suppose f is an elementary diffeomorphisra and x is a quasi-tra~ersal intersection of W~(p,f) and W'(q,f) with p and q periodicpoints off. Then f is not topologically conjugate to any Kupka-~Tmale diffeomorphism. Moreover, if f' is also elementary and h is a topological conjugacy between f and f', then h (x) is a quasi-transversal intersection of hW~(p, f) and hW'(q, f). Proof. ~ This follows from the fact that a transversal intersection of two submani- folds W' and W" of a manifold M is topologically different from any quasi-transversal intersection of W' and W ". In a point of quasi-transversal intersection, either dim W' -4- dim W ~ 4= dim M 36 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 37 in which case the intersection is not a manifold of the right dimension or, dim W ~ + dim W ~ = dim M in which case the intersection number is zero. See Chapter II, section 6. Note that not all non-transversal intersections are topologically different from transversal ones. Theorem (*.3)- -- Suppose f, p, q, ~, f~, and r are as in the hypotheses of Theorem (1. x). Then there is a residual set ~t C ~ such that if ~ e ~x, then no %, o <__ ~ <__ i, is topologically conjugate to dr. Proof. ~ There is a residual set 9~0 C 9 ~ such that if $ e 9~0, then (i) each %, is elementary or Kupka-Smale; (2) if some %, has a quasi-transversal orbit, then the hypotheses of Theorem (,. i) with %, =f' are satisfied; (3) there are at most countably many lz's for which %, has a quasi-transversal orbit. Now, since each % has at most one quasi-transversal orbit, it follows that the quotients of the logarithms of the moduli of quasi-transversal orbits occurring in {%} form a set which is at most countable. With standard arguments, one can show that there is a residual subset ~1 of 9~0 such that if 9 z t~x, then all the quotients occurring for ~ are different from log [0c] Now Theorem (,. 3) follows from Theorem (,. x) and Lemma (I. 2). log [ ~ [" Corollary (*.4). -- If ~ e ~ is left stable, then all stable andunstable manifolds of periodic points of 9b intersect transversally. Proof. -- Since q~ e ~, we have that % is elementary. Thus, if % =f has a quasi-transversal orbit, then we may assume the hypotheses of Theorem (i. i) and hence Theorem (i .3) are satisfied. By Theorem (I .3), q~ is not left stable. 2. Necessary conditions for mild stability and stability Suppose q~ e ~ and ~(p) is a saddle-node for %. We say ~(p) is s-critical for % if there is a periodic orbit r such that W~'(~(q)) has a non-transversal intersection with the strong stable foliation o ~-s' of W~(~(p)). Similarly, we say ~(p) is u-critical if it is s-critical for q~-l. If ~(p) is either s-critical or u-critical but not both, we say that ~(p) is semi-critical. If ~(p) is both s-critical and u-critical, we say it is bicritical. Proposition (2. ,). -- If ~ is mildly stable, then it is left stable and the quasi-hyperbolic periodic orbit is not a Hopf orbit or bi.critical saddle-node. 8~ 3 8 S. NEWHOUSE, J. PALIS AND F. TAKENS Proof. -- The first statement is obvious and the second statement was proved in Section 5 of Chapter II. Now suppose r is a bicritical saddle-node for ~b" We assume p is a fixed point of ~b" If qx and q, are periodic points of % such that W"(qx) has a non-transverse intersection with the strong stable foliation of W'(p) and W'(q,) has a non-transverse intersection with the strong unstable foliation of W"(p), then, perturbing if necessary, we may assume that these intersections are quasi-transverse. For some ~ near b, it then follows that W"(qa~, q~) has a quasi-transverse intersection with W'(q~,, q~). Here, of course, qx~ and q~ denote the unique hyperbolic periodic points of ~ near ql and qz, respectively. Using Theorem (1.3) we can perturb to ~' in ~ so that no q~', is conjugate to %,. This shows that ~ is not mildly stable. Proposition (2.2). -- If 9 is mildly stable, then ~b has no cycle of length bigger than one. Proof. -- Suppose q~b has a cycle of length bigger than one. We show ~ is not mildly stable. Since q~ ~ ~, q~b has a saddle-node orbit *(Po) which is contained in every cycle. Indeed, the transversality of the stable and unstable manifolds implies that if 9b had any cycle not containing a saddle nodes, then ~b would have transversal homo- clinic points. This would give 90 infinitely many periodic points [34]. Let r 9 9 -, r be the distinct orbits in the cycles of q~b- Replacing { q~ } by some power { ~, }, we assume that all the p+'s are fixed points. Let W"(P0), resp. W"(p0) , denote the strong stable, resp. unstable, manifold of Po. Since L(~b) has finitely many orbits it follows that W"(p~) n W'8(po) = o and W'(p+) n W"(p0) = o for 0 < i < r. Otherwise, q~b would again have transversal homoclinic points. By transversality, we have dim W"(p+) = dim W"(po) or dim W"(p+) = dim W"(po) -- i for each o < i < r. Replacing {~} by {~-1}, if necessary, it is enough to consider the case in which there is a I <j < r such that dimW"(p~) = dimW"(Po). This implies dim W*(pj) n W"(Po) = o and hence Po is u-critical. Observe that, having dim W"(pj) = dim W"(P0) for some i < j < r, we may indeed assume that for all o < i < r (a) dim W~(p+) = dim W"(Po) , (b) W~'(p+) is transverse to the strong stable foliation ~-" of W'(Po), (c) e w"(p+) w"(po) = {Po}. For suppose (a) or (b) failed for some I < i < r. Then Po is s-critical. Since dim W"(pj) = dim W"(Po) , we already know that Po is u-critical. Thus Po is bicritical and Proposition (2. I) has already ruled this out. Let us now suppose that (b) fails for i = o; that is, W"(P0) has a nontrartsverse intersection with .~'". We already know that W"(p~) is transverse to ~" for I < i < r. Perturbing % if necessary, we may assume that W"(p0) has a quasi-transversal intersection with some leaf F in ~". Choose disks Dx C W"(P0) and D 2 C F so that D x and D2 have a quasi-transversal intersection. Let 38 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS be an invariant center manifold in W'(P0). Since W"(pt, b) meets W'(po) transversally. and is transverse to o~', it follows that each component of W~(px, b) n W'(P0) is an embedded curve whose closure near P0 can be given as the graph of a smooth function from ~ to W"(po) as in the next figure. n WS (Po) ~ ~~ Fro. 6 As ~t increases, pieces of W"(Pl, ~t) will sweep across the local part of W'~o ) near Po, so for each Ez > b near b, W"(px, ~t) contains a disk Dx,~G ~ near D1. On the other hand, W'~x , b) accumulates backward on W"(p0). For certain tz's near b and greater than b, W'(pt, bt) will contain a disk D2,~C r near D,. Considering the continuous movements of Dx,~ and D2.~, as ~ varies, one sees that for certain ~t's near b and greater than b, Dx,~ has a quasi-transverse intersection with D~,~. Thus, if (b) fails for i = o, there are ~t's near b for which W"(pl,~) has a quasi-transverse intersection with W'(Pt,L,)- In view of Theorem (x-3), we conclude that ? is not mildly stable. Finally if (c) failed for some i, then d W"(p~) would meet a fundamental domain for W'(p0). Since all intersections of W"(A) and W'(pk) are transverse, this implies W"(p~) n W"(p0) . ~ as in the Morse-Smale case ([24]; Lemma (i.5)). As we have already mentioned, this gives transverse homoclinic points--an impossiblity. Thus, we assume (a), (b), and (c) hold for all o < i < r. We proceed to derive a contradiction from this, and, then, Proposition (2.2) will be proved. From (a), (b), and (c) it follows that each component of W"(pi) n W'(P0), o < i < r, is an embedded curve whose closure near P0 is the graph of a smooth function from the center manifold ~ to W"(p0). Let Pi, be a fixed point such that W~(p~.) n W'(P0)  ~, W"(Po) n W'(A.) 4= o, and for each x <i<r with i+io, W"(p~.) nW'(A ) =o. Then, W"(pl.) nW'(P0) consists of finitely many curves which are permuted by ~Pb. Also, the boundary of each suchcurveis{po,p~,}. By the boundary of a curve-(, we meand~, -- y, ofcourse. Since W"(p~.) is transverse to W'(po) , we conclude that the closure of W'(po) near Pi. is a finite union of closed half-spaces bounded by W'(p~.). Also, since W"(po) has a transverse intersection with W'(pi, ) and dim W"(po) = dim W"(p~,), W"(po) accumulates on W"(pl,) in the G I sense. Thus, we see that W"(P0) n W'(po) has a component T which WU(pl) 40 S. NEWHOUSE, J. PALIS AND F. TAKENS is an embedded curve whose boundary is {P0, x} where x e Ws(p~.) -- {pl.} n W"(Po). This is depicted in the next figure. Fxo. 7 We will show that the existence of T leads to a contradiction. Let D be a small half- disk neighborhood about Po in W"(Po) so that %(OD -- W"(po) ) n (0D -- W~(po)) ---- o. Let D" = cg(~bD -- D) be a fundamental domain for W"(p0). Choose N > o so that n ~ N implies that ~;-"x ~ D. It follows that, for n > N, %-"? is an arc whose intersection with D" -- W"(p0) has at least one component joining the two components of O(D" -- W~'(p0)). We may suppose that 0D" is transversal to each curve ~gy, r E Z. Hence, the set E of points y ~ D" -- W~(p0), for which there is a sequence Yl ~ ?;-,i(y), ni ~ oo, such that yi -+y, is uncountable. But E Cct Ws(p0) n D". We assert that E n W'(P0) ----- o. For the moment, assume this assertion holds. Then E C [.J W'(p~). Since dim W'(pl) + dim W"(P0) = dim M, this would put E a<i_<r in the countable set D ~ n [3 WS(pi) ) which is a contradiction. l<i<r Thus, we must prove that V, n W'(po) ---- o. Suppose E n W'(po) ~ o and let y ~ E n W'(P0). Choose y~ E q~b-"i(T), ni -+ 0% so that yi --~.y as i -+ oo. We claim that yi goes to infinity in WS(po) ; i.e. given any closed disc F C W'(P0) transverse to D" there is an i o~'o so that .y~r for i>i 0. This follows from the facts that 9~'~, n 9;V---- ~ for m~e r and F n D" has only finitely many components. Now let D' and Ds' be fundamental domains for W'(p0) and W"(po), respectively, and let D "~ be a fundamental domain for W~'(po). There is an N ~> o so that ~?~(y) e D ". Since 9~(y~) converges to the point ~(y) in D s and %s(y~) goes to infinity in WS(po), there is an m~  o such that ?~+mi(yi) accumulates on D ~". Thus, d W'(po) n D ~" ~e o. But this implies that % has transversal homoclinic points which is impossible. Thus E n W'(P0) ---- o and Proposition (2.2) is proved. Proposition (~. 3). ~ If ~ is mildly stable, then ~b cannot have a non-critical i-cycle. Proof. ~ As before, if % has such a cycle, it must have a saddle-node ~(p) which is in the cycle. Since the cycle is non-critical, W*(r is transverse to the strong stable 40 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 4 z foliation of W'(r(p)) and W*(o(p)) is transverse to the strong unstable foliation of W"(,(p)). Let nx be the period ofp. Using invariant manifold theory one sees that W"(~(p)) n W'(~(p)) is a finite union of C ~ circles Slb, ..., Skb which are permuted by 9~ '. Taking n i larger, we may assume ~b (Sib) S~ for I < i < k. Replacing 9 by a suitable C ~176 approximation, we may assume for ~ near b that there is a C ~~ circle S~g near Slb which is invariant by q~', and the rotation number of 9~ ' [ S~ is not constant as a function of ~ near b. From the recent work of M. Herman [9] we may find a near b such that 9~,'[ Sl~ is C oo conjugate to a geometric rotation through an angle with 0~12~ irrational. By a small modification of r to q~', we may make ~p~"' [ Sl~C ~~ conjugate to a rotation through an angle a' with 0c'/2~ rational. This means that %, has uncountably many periodic points of the same period. But, since each r is elementary, it has only finitely many periodic points of a given period. Thus ~ is not mildly stable. Remark. -- The previous propositions leave open the possibility that a mildly stable arc might have a critical saddle-node in a t-cycle at %. We will wove in section 4 that this cannot occur if the stable or unstable manifold of the saddle-node is one dimensional. In particular, it cannot occur if dim M = ~. In general, for dim M > 2, we feel that this cannot occur, but we have no complete woof at this time. The next proposition shows that the situation regarding stability is better. Proposition (2.4). -- If q~ is stable, then 9b has no cycles and no semi-critical saddle-nodes. Proof. -- We have already taken care of cycles of length bigger than one and non-critical i-cycles in Propositions (2.2) and (2.3). Suppose ~b has a semi-critical z-cycle containing the saddle-node orbit ~(p). We assume p is fixed by % and W"(p) has a non-transverse intersection with the strong stable foliation .~* of W'(p). The other cases are handled similarly. Perturbing, if necessary, we may assume that o ~'~' is a C 2 foliation, that all intersections of W"(p) with leaves of o ~-"* are transverse or quasi- transverse, and that all the eigenvalues of d% on T~W'(p) have multiplicity one with distinct norms. Let 0c be the weakest contracting eigenvalue of d% on TvW*'(p), and let Cp be a C t invariant manifold in W"(p) tangent at p to the eigenspace of a. Let Hp be a C t invariant manifold containing W"(p) and Cp as in the woof of Theorem (t. i). Let x e W"(p) n W'(p) be a quasi-transversal intersection of W"(p) and ~-~' at x. If y is a curve in Hp n W*(p) transverse to W"(p) at x, then Y is transverse to o~-~' in WS(p) at x and, hence, projects diffeomorphicaUy near x along the leaves of o~-" into an invariant center manifold ~ for %. Let D be a fundamental domain for W"(p). Thus D is a compact set in W"(p) such that 9~D nD = o and W"(p) --{p}C [.J ~](D). Let n~Z U be a small compact neighborhood of D. If y e 7 is near x, then O(Y, x) ,~ [ ~ I "Iv) where n(y) is the least positive integer n such that q~-"(y) ~ U. Here, as in the proof ofTheorem (z. t), we use p(y, x) -~ [a[" to mean that ~ is bounded and bounded away from zero independent of n. 41 S. NEWHOUSE, J. PALIS AND F. TAKENS 4 ~ Now let 9' be a perturbation of 9 such that b(q~') ---- b(~), [a'[ < [0~l, and ~'~ is still a C 2 foliation. Using Chapter II, we may assume that on the center manifold through p, 9b is the time-one map of a C | vector field X b which vanishes only at p. Also, there is a corresponding vector field X; for ~. Suppose there is a continuous conjugacy { h~ } between q~ and ~'. By Theorems (3.2) and (3.4) of Chapter II, we know that (I) h b is C ~~ along ~--{p}; (~) h b maps X b to X;; and (3) hb maps the strong stable foliation ~'*' to the corresponding strong stable folia- tion o ~'u. Since ~'"' is C 2 near ~", all intersections of W"(p') and leaves of ~'"8 will be transversal or quasi-transversal. As in the proof of Lemma (i. 2), this implies that hbX is a quasi-transversal intersection of W"(p ') and o~-;,b,. '" Applying the reasoning in section D of the proof of Theorem (i. i) to W~"(p), we see that, since H r is transverse to ~'~', the invariant manifold in W88(p) tangent to the sum of the eigenspaces complementary to the eigenspace of 0r has a dynamical characterization. It is the set of points where the backward orbit of~'~ ' accumulates. Thus, hb(H p n W"(p)) is in a sector about H~ c~ W"*(p ') in W'8(p'). This implies that ify is near x in y and p(x,y)--~[a[ "lu), then p(h~, hbX ) ~ 10r Hence, (4) lim p(hby' hnx) -- o. v-~ p(y, x) If r~ : W*(p) ~ ~ is the projection along the leaves of ~'b", then (4) implies that hb has derivative zero at ~(x). But this contradicts the fact that hb(Xb(~x)) ---- X'b(hb~(X)). The proof of the fact that stability of ~ implies that % has no saddle-node orbit r which is semi-critical via some other periodic orbit r is similar. If say W*(r has a non-transverse intersection with o~'~', we repeat the preceding argument replacing WS*(r by WS(r The remaining case of W*(~(q)) having a non-transverse inter- section with ~'~" follows replacing { %} by { ~- 1}. Remark. -- We observe that stable arcs in d must lie in ~. This follows from a somewhat more general observation, namely that any stable arc {q~}, not necessarily starting in MS, with a bifurcation for bt = b such that the limit set of ~b has finitely many orbits has the following properties: -- all stable, strong stable, unstable, and strong unstable manifolds of ~b intersect transversally; -- % has exactly one non-hyperbolic periodic orbit, which is either a flip or a non-critical saddle-node without a cycle; this non-hyperbolic periodic orbit unfolds generically; such a ~b can have no cycles because it has finitely many orbits in its limit set; -- there is an r such that if ~ c (b -- r b q- r -- {b}, then %, is MS. 42 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS This statement follows from the previous arguments in the following way. If all the periodic orbits of % are hyperbofic then, since the limit set has only finitely many orbits and % is not structurally stable, there must be a non-transversal intersection of a stable and an unstable manifold. By the results of section one of this chapter the arc { %,} would then be unstable. Also, the non-hyperbolic periodic points have to unfold generically otherwise we would not even have local stability. It is also clear that no more than one orbit of % can be non-hyperbolic. If % has a Hopf point or a non-critical I-cycle, then there are nearby arcs with a smooth invariant circle with irrational rotation; as observed in the present section that also contradicts stability of the arc. Also in this section we saw that a saddle-node with criticaUity and or a cycle of length bigger than one is impossible if the arc { %,} is stable. If ~b has a non-transverse intersection of the stable, strong stable, unstable, or strong unstable manifold of the non-hyperbolic periodic point with any of the other stable or unstable manifolds, then there is arbitrarily near { ~} an arc { %,} and a ~ ~ R near b such that ~ has a non-transversal intersection of a stable and an unstable manifold of hyperbolic periodic orbits. Again by the results of section I this implies {r and hence, {~} is unstable. Finally, we indicate why q~ r MS for near b and tz # b. Since % has no cycles and the limit set of % has only finitely many orbits, the Birkhoff center of Sb must be finite. Then the arguments in [i5] show that the limit set of q~b equals the Birkhoff center, and, hence, must also be finite. Because Sa is elementary, it then follows that q~, ~ MS for ~ near b and V # b. With arguments similar to those in the preceding paragraph, one can show that a mildly stable arc in d must lie in ~. There are, however, left stable arcs in .~r which are not in ~g: % might, for example, have two Hopf points or flips. All such arcs are left conjugate to left stable arcs in ~. 3" Endomorphlsms of the drcle In this section we first extend the notion of rotation number, defined by H. Poincar6 for diffeomorphisms of the circle, to endomorphisms of degree one. Instead of a number we get in general a closed interval of the real line, which we call rotation set. These rotation sets are then used to analyze a class of I-parameter families of endomorphisms with non-degenerate folds (see (b) below). We show that each such family must go through homoclinic trajectories with folds (see Theorem (3.7)). This result has a direct application to the bifurcation of diffeomorphisms exhibiting a saddle-node with one cycle described in the previous section. If the saddle-node has a one dimensional stable or unstable manifold, they must go through a non-transversal homoclinic orbit. An interesting question is if such families of endomorphisms, for which the rotation sets vary, have bifurcation sets of positive measure. For the diffeomorphism case, see [IO]. A) Rotation sets for endomorphisms of the circle We identify the circle S 1 with R/Z. By End(S) we denote the set of continuous maps 9 : S 1 ~ S 1 of degree i. On End(S 1) we use the usual C o topology. For each 43 S. NEWHOUSE, J. PALIS AND F. TAKENS eEnd(S t ) there is a ~:it-+it such that r~=~ where 7=:it--->S t--->R/z is the canonical projection. ~ is called a lift of 9 and it is unique up to the addition of an integer. Each such ~li satisfies ~(x + i) = ~(x) + I. For 9 t End(S t) with lift ~ and x ~ R we define the rotation number p(~, x) to be lim sup-X (~"(x) --x) and the rotation set p(~) to be closure {p(~, x)[x eit}. n~o n Note that if we take a different lift, say ~, of 9 then p(~, x) and p(~, x), and hence also p(~) and p(~), are equal up to translation by some integer. If x' ~ It with ~(x') = ~(x) then p(~, x') = p(~, x). Hence, up to translation by integers, p(~, x) and p(~) are invariants of ~, ~(x); if no confusion seems possible we may denote them by p(r ~(x)), O(~). We note that p(~, p) and p(~), p ~ S t, are topological invariants: if h : S t ~ S t is a homeomorphism, then O((1),p) ---- p(h*h -t, h(p)) and p((l)) = p(h(I)h-X). Finally, if q) is an orientation preserving homeomorphism, then p((I)) is the usual rotation number [29]. Lemma (3. 9 ). -- If dp t End(S t) and ~) is a lift of r and if there is no periodic point of r with rotation number P-, p e Z, q t N, i.e. if there is no x t It with r = ~(x) and p(~,x) P, then p(~) is contained in { x t it , x < P } or in {xtR[x>P} Proof. ~ If, for some x e It, ~(x) -- x = p then ~(x) is a periodic point with rotation number -P- ; so this does not happen. Hence either ~q(x) -- x < p for all x t It or ~q(x)- x > p for all x e It. Since ~q(x)- x is periodic in x (with period I) there is some ~ > o such that ~(x) -- x < p -- r for all x e R or ~(x) -- x > p -k- r for -- q q Corollary (3.2). -- If 9 tEnd(St), a, ~ tp(~) and ~<P-< ~ for some rational number p, then 9 has a periodic point with rotation number p-q and hence P-q ~ p(~). Since p(~) is closed (by definition), p (~) must be either a single point in It or a closed interval. Now we want to show that p(*) depends continuously on (1). For this we introduce the following notation: if 9 t End(St), ~ a lifting of~ then pl(~), p~(~) t It are such that p(~) = [p~(~), p~(~)]. Proposition (3.3). -- Let UCEnd(S 1) be some open set such that there is a continuous mapping 9 -+ ~ which assigns to each 9 a lifting r (in order that ~ depends continuously 44 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS on ~, U should not be too big). Then the functions gP ~ px(~li) and ~ w, p~(~) on U are cont/nuous Proof. -- We observe that for any rational number P, we have that P- q < pt(~) is equivalent with ~q(x) -- x :> p for all x e [o, x] which is an " open condition ", i.e. the set of 9 E U with pt(~)~P- is open. Analogously, the set of those 9 e U with pa(~) ( P- is open. e (pl(~), pa(~)) if and only if for some big N e N, there are x,y E [o, i] Finally, p with ~sq(x) --x~> N.p + i and ~'q(),) --y( N.p-- i. Also this condition is open, hence pl(~) and ~(~J) depend continuously on ~. Proposition (3-4). ~ Let r e End(S t) with lifting ~ and let P- e (01(~), [~1(~)). Then there is a periodic point s ~ S t of 9 with rotation number P-- such that W~(s) = S x, where Wg(s) --= 9 [U r the intersection being taken over neighborhoods U of s. "EN Proof. ~ We may assume that P- = o (if not we replace 9 by r and choose an appropriate lifting). Let X denote the projection of the fixed point set of ~ in St; since X is closed and non-empty, S ~- X consists of open intervals. Let U be such an interval and U = (~-t, ~) a lifting of U. Then ~(x) -- x, for x ~ U, is always positive or always negative. In the first case, U C W~(n(s-t)), in the second case, U C W~(=(~)). If 9 ~"(U) =S ~ we may take s=r~(s-1) or s=~(s~) and are done. If on the other n 0 hand for each component U of S 1 -- X, ,>~0~"(U) + S t, then r x]) C [-- ~, 2] for all n>o, so we would clearly have p(r ={o}. This would be against the assumptions. Remark (3.5). ~ Let ~ C End(S t) denote the subset of those 9 e End(S 1) which are G t, whose first derivative is of bounded variation and whose critical set consists of a finite and non-zero number of generic folds, i.e. points where 9 is locally C ~ conjugate with the map y = x ~. By a theorem of Block and Franke [2], every 9 in ~ has a periodic point. This is equivalent with saying that if 9 E ~, p(cP) cannot consist of only one irrational real number. This together with the continuity of Px and p~ implies that if R ~ ~ ~-~ r is a continuous arc in End(S t) with image in aj then p(~o) is constant (as function of or) or, for some ~, p(O~) has a non-empty interior. This observation will be useful in the next section. Finally, we pose the following conjecture which extends Herman's result for arcs of diffeomorphisms [i o]. Let { q~} be a C t arc of C ~ endomorphisms of S 1 with rotation sets { [Pt(D), P2(g-)] }- 45 s. NEWHOUSE, J. PALIS AND F. TAKENS Conjecture (3. fi ). -- If the rotation set of r varies with tz, then the set of t~'s for which either Pt(~) or P~(t~) is irrational has positive (Lebesgue) measure. B) Some one-parameter families of endomorphisms of S 1 In this sub-section we consider continuous (or smooth) families ~o, a z IR, of endomorphisms of S t such that for each ~ ~ R, r E ~, see Remark (3.5); the rotation set p(~o) is non-constant, as a function of a; -- for each rational number P, there is a locally finite set Zp C R such that for a r Zp, all periodic points of ~o with rotation number p- are hyperbolic. We shall prove for these families: Theorem (3.7). -- For % as above there are an interval (6t, %) C R, a hyperbolic periodic point s o , a t < a< % of ~o such that W~o(so) is S t for all a ~ (at, %) and a point to, a t < ~ < ~ in the critical image of r (for some n) such that: both s o and t o depend continuously on 6; the curves ~ ~ (so, a) and a ~ (to, a) in S t � (at, a,) cross one another; if ~o is a lifting of ~o, depending continuously on a, and if so is a lifting of so, then t o is the projection of an end point of the interval ~(~o,-so + i ). Remark (3.8). -- For 9 e End(S1), with lifting ~ one can construct a I-parameter family of endomorphisrns q)o by putting q)o(=(x)) = ~(~(x + a)). We note that in the set of (3' endomorphisms ~ of S t, r > 2, which are not diffeomorphisms, there is a residual subset for which the corresponding one parameter families of endomorphisms satisfy the assumptions in Theorem (3.7). Proof. ~ Since p(~o) is non-constant, there is some a3 such that p(~o,) has interior points, see (3.5), and such that p(~o) is not locally constant on a neighborhood of %. Choose a rational number P- in the interior of the rotation set. Since in any neighborhood of % there are infinitely many points where the function p(~o) is not locally constant, we can choose at such that ~o, has a hyperbolic periodic point s with rotation number P- such that W~o,(S) = S t (see (3.4)), and such that p(~o) is not locally constant at at. Now if follows that there is some neighborhood U of a4 in R and a continuous function U ~ a ~ so e S t such that, for a E U, so is a hyperbolic periodic point and such that W~o(So) = S t. Next, since p(~o) is not locally constant at r there is a ~ in U for which p(q~o,) # p(q~). Then one can find two rational numbers Pt P2 with Pt < a0~ such that 46 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS Pl P2] the closed interval ~, ~ is in (O(r -- p(~)) v (p(~) -- p(~o,)). We suppose ~,~ is in 0(~o,) -- P(q~) since the other case is similar. Then, with glio, ~-o liftings of q)o, So, + + = + 3Pl, + 3P ] and _~ ~, ~-~ + x) C complement of [s-~ + I + 3Px, s-~ + 3Pa] ; see also Lemma (3.1). This means that one of the endpoints of the interval O~(s-o, ~-o + i), cr between ~ and ~, has to cross over the interval [~-o + I + 3Pl, To + 3P~]. Since this last interval has length at least 2, one of the endpoints of O~'(~-o, ~-o + i) crosses, after projection on S ~, over So when ~r goes from ~ to 7. Finally we have to show that if, for ~ between cr~ and 7, one endpoint of O~'(~-o, ~-o + x) lies in [~-o + I + 3Pl, To + 3P~] then this endpoint is in the critical image of =o~S~'. To show this, it is enough to show that neither ~'(~-o) nor O~'(~-o + I) can be in ~-o + i + 3Pa, So + 3Ps]- But this follows from the fact that the rotation number of So is P- and that P- does not lie between P x and p~ (because otherwise P- r p(Oo), see (3-~)). q q qx ql q Now it is clear that we can choose (~, ~,) to be an interval between e~ and ~ such that if ~ goes from ~x to ~, one endpoint of ~'(~-o, S-o+X) crosses over the interval [~-o + x + 3Pt, To + 3Pa]. We choose then n = 3ql and we choose to to be the pro- jection of that endpoint. C) Application to bifurcation theory The analysis of families of endomorphisms in the last section leads to results also when analyzing certain families of diffeomorphisms. An explicit formulation of what we need in this direction is: Remark (3-9). -- Let ~o : S 1 ~ S l be a one-parameter family (r e R) of endo- morphisms as in Theorem (3-7)- Let ~o,~ : S 1 � [o, i] ~ S 1 � [o, i] be a 2-parameter family of C ~ maps depending continuously on (a, ~) such that: -- q)o,0(x, t) ---- (q~o(X), To(x)) for some function To; -- for ~z positive, 0o. ~ is a diffeomorphism into. Then, for ~ sufficiently small, there is a a~ E (el, ~,) such that *~g.~ has a non. transversal homoclinic point. Even more holds: for ~ sufficiently small, and any curve (a,f(,)) with o <f(a) < ~, e (*l, as), there is a ~/such that r ) has a non-transversal homoclinic point. The proof of this statement follows from the following continuity considerations. The map q)o.~ has a hyperbolic periodic point so,~ = (Xo.~, to,~) such that Xo, 0 = so 47 S. NEWHOUSE, J. PALIS AND F. TAKENS as in Theorem (3.7). We observe that W'(so,~,) depends continuously on (~, [z) and W*(s,.0) = {Xo,0} � [o, I] (note that % : S 1 -~S 1 was expanding at So). One can choose a local unstable manifold W~o0(So.~) as an embedding (or its image) which depends continuously on (~, ~). For some k, k ?o,0(Wl0~(Xo,0) ) has a point To in its critical image which moves over So, 0 by Theorem (3.7). From this and the above continuity we see that, for each small enough tz > o, there is a value of ~ such that k is not transversal with respect to W"(so,~). 4. The saddle-node with x-cycle in dimension 2 Let {?~}: M -+ M be an arc of diffeomorphisms of a compact 2-manifold M so that for ~ < b, % is Morse-Smale and such that for [z = b, % has a saddle-node at p (we assume p to be a fixed point of % ; the case where p is periodic can be handled analo- gously). Further we assume that % has a x-cycle containing p; i.e. there are non-trivial intersections of W"(p) with W'(p) and there is no periodic point q such that both W~(p) ~ W'(q) and W~(q) n W'(p) are non-empty. Theorem (4. 9 -- Under the above hypotheses { % } is not mildly stable. Remark (4.2). -- It will be evident from the proof of Theorem (4. i) that the same result holds if dim 1V[ > 2 and % has a saddle-node in a l-cycle whose stable or unstable manifold is one dimensional; i.e. a normally repelling or attracting saddle-node. An open dense set of these arcs will create homoclinic tangencies. Thus, in dimension two they will contain diffeomorphisms with infinitely many sinks or sources [2~]. Proof of Theorem (4. l ). -- An arc { %} is already not mildly-stable if in any neighbor- hood of % there is a non mildly-stable arc. Hence we may, without loss of generality, impose generic conditions on { % }. In particular, we assume that p unfolds generically. Also, we may assume that the eigenvalues of (d%)v are i and ~r with o < ~ < i ; in case 0t> I we take?~-l, incase -- i< 0t<o we take ?~ and in case 0t<-- I we takerS2 instead of 9. From the fact that, for ~ < b, ?~, is Morse-Smale and the fact that {%,} unfolds generically at p, we may assume that, near p, ?~ has two fixed points for tz < b, one fixed point for ~z ----- b and no fixed points for ~z > b; we also assume that W~(P, ~b) n 0(W'(p, ~b)) = {p}. We first observe that W"(p, %) C W'(p, %); this follows from the following two facts: -- W"(p, %) -- {p} t~ W'(p, %) is open in W~(p, %) -- {p} and non-empty; -- for any filtration {M~} of M for ~b (see [20]), with p e Mi. -- M~._I, the set of points in W"(p, ?b)- {P}, which go eventually into Mi_ 1 is open and equals (w-(p, -- {P }) -- W'(p, 48 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS In the case that W~(p, %) closes to a smooth circle, p has a non-critical I-cycle. Then the instability here was proved in Proposition (2.3). Thus, we may suppose W~(p, %) does not close to a smooth circle. This means that the projection of W~'(p, %) on a center manifold in W'(p, %) along the strong stable foliation does not have maximal rank everywhere. We shall approximate { %, } by a family {~,} such that non-transversal intersections of stable and unstable manifolds occur for certain ~'s arbitrarily near the first bifurcation of{~'~}. From this, we infer that {~"~}, and, hence also {%,}, is not mildly stable. This will prove Theorem (4. I). For convenience of notation, let us assume that { %, } is defined for Ez near o and that Ez = o is the bifurcation point instead of ~ = b. For each diffeo- morphism ~'~ we require that there is a smooth vector field X~, defined for ~z close to o on a neighborhood of p, such that the time x map X~. x of X~, wherever defined, equals ~. Also we require that there are smooth coordinates y, z (which may depend on ~) such that X~, locally has the form 0 0 X~ = Y~(y) ~y + Z,(y).z. Oz O~Y~176 OY ~ > o, and { Y~,(y) ~y } where Y and Z are smooth functions of (y, Ez), Zo(o) < o, is a saddle-node arc. The fact that such ~"~ exists arbitrarily close to (p~ follows from [38], [39]. Choose, for ~ > o a fundamental domain D2.~ for ~ in the positive y-axis, smoothly depending on F~. For some big m, ~'~(D2,0) will again be in our coordinate ~o ~ O~(D~. o) ,~, ', D ,o D2,0 z< Flo. 8 S. NEWHOUSE, J. PALIS AND F. TAKENS 5o neighborhood the same holds for ~"(D2,N) if ~x is near o. Define D1, N to be the funda- mental domain in the negativey-axis whose boundary is the projection, along the z-direction, of the boundary ~" of %, (D2,N). We take a " rectangle " ~N SO that its boundary consists of two pieces of XN-integral curves and two pieces of straight lines parallel to the z-axis, such that its projection on they-axis is D1,N, and such that for each q e ~'~(D2,N) , there is a positive m' such that ~'(q) is in the forward orbit of 9,,. See figure 8. Let A N : 9N "--> M, ~ > o be defined as follows: Ao(~o) = D2,o and Ao(y , z) = (ao(y), o) where a 0:Dx, o~D2'0 satisfies (ao).X o=X o on {z=o}. For ~>o there is a positive T N such that the time T N map of XN, (XN)~N, satisfies (XN)~ND1, ~ = D~,~; we define A N to be (XN)r. For e e [o, I] we define Ao,~ ----= (XN) o oA~,. Next we define Bo, N ----~' o Ao,~. For ~ close enough to o, the image of Bo,~ will be in our coordinate neighborhood in the part {y < o }, and also there is some posi- ~.,, tive m' such that the image of ~ o Bo,~ is in the forward orbit of ~N" Consider now the quotient map {(y, z) [y< o} -+ {(y, z)[y< o}/(y, z) '~ ~N(Y, z). Under this quotient, ~, becomes an annulus ~, Bo, N becomes a z-parameter family of mappings Bo, N of ~N into itself and the projection ~(y, z) = (y, o) goes over into ~, the annulus projection. We want to show that Remark (3.9) is applicable to BO, N" In order to prove this we need new coordinate functions s~:~ N -+ R/Z and w~, : ~N ~ IR such that ~(sN, w~,) = (s~,, o) and such that X~, has the form --. We Os N shall write s, w instead of s0, w 0. For some circle endomorphism ~ : R/Z -+ R/Z and some map W : R/Z ~R we have B~ w) = (~(s -k ~), W(s)). The circle endo- morphism ~ is determined by ~'0; ~'0 was obtained from % by a small, but otherwise arbitrary, perturbation. Hence we may, and do, assume that the family { ~o } defined by ~o(s) = ~(s + ~) satisfies the assumptions of Theorem (3.7); see also Remark (3.8). Now it is clear that Bo, N satisfies the assumptions in Remark (3.9). Now we observe that if T~, q- ~ eN, BO.N is an iterate of~, (up to the identifi- cations). Hence if such a Bo,~ has a non-generic tangency of a stable and an unstable manifold then the same holds for ~'N" Also, Bo,~ satisfies the conditions of ~o,~, in Remark (3-9). Let (~i, ~2) and ~ be as in that remark. Now as ~t -+o, we have T~, -+ oo. For each a ~ (~ri, ~) choose anf(~) such that TI(o) + cre N, and f(a) < "~. We may arrange for a ~f(~) to be continuous on (cl, ~2). Then as in Remark (3.9), there is a at ~ (at, ~,) so that Bot.t(otl , and hence ~~ has a non-transversal homo- clinic point. 50 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 5" On the rigidity of the nnfolrl;ng of the saddle-node In Chapter II we have seen that a conjugacy between two Coo saddle-node bifur- cations must satisfy very restrictive conditions. In particular, at the bifurcation parameter it must be C Oo along central curves away from the fixed point, and it must preserve adapted saddle-node vector fields along these curves. In the present section, we shall present two more applications of the restrictive nature of such conjugacies. The first concerns certain arcs of diffeomorphisms of the circle, and the second concerns arcs between Anosov diffeomorphisms and so-called DA diffeomorphisms. These furnish more examples where mild conjugacies cannot be strengthened to conjugacies. A) One-parameter families of diffeomorphisms of S 1 We consider one-parameter families q~:S1---> S 1 of Coo-diffeomorphisms of S 1 with rotation number p(~t) increasing such that whenever 0(5) is rational, q~g has two hyperbolic periodic points or one periodic point of saddle-node type which unfolds generically and such that whenever p(~) is irrational, p is not locally constant in ~. Let ~'~, : S I --> S 1 be another such family of diffeomorphisms with rotation number p'(~). If we assume that Image(p) = Image(9'), then there is a homeomorphism h : R -+ It such that p'(h(~)) = p(~t), see [5], [IO]. For each ~, r and q~,) are now conjugate: for p(~t) irrational this is Denjoy's theorem [6], for p(~t) rational it follows from the above description. So it is clear that the above two arcs are " mildly conjugate " A question, which was raised independently by R. Thorn and S. Smale, is: are two arcs as above in general conjugate? We show here that the answer is in general negative. Let {%,}, {~0~} be a pair of arcs of diffeomorphisms of S 1 as above. Let ~. be a boundary point of p- 1 (~) for some rational P- q e Image(p). If there is a conjugacy (h, H~) between % and %, i.e. h : It ~ It a homeomorphism and H~, a conjugacy between % /L't i ouo y o ntof,, .ow and ~)depending continuously on ~t, then h(~) q~ has a unique fixed point which is of saddle-node type. By Chapter II, Theorem (3. ~), there is a unique smooth vector field X near the saddle-node orbit such that its time one map X x equals (~g)q (and such that (~g).~X = X). Also for q~h(~,l there is such a vector field X'. Again by Chapter II, Theorem (3.2), Hg has to map X to X'. If we now extend X, and X', to all of S t so that ~g.X = X, q~(~l.X' = X', we obtain in general bivalued vector fields. This means that, on the complement of the periodic orbit, Hg has to respect two different vector fields. This is in general impossible. B) Arcs between Anosov and DA diffeomorphisms In [34], S. Smale showed that certain Anosov diffeomorphisms may be modified to give Axiom A diffeomorphisms with attractors having intricate topological properties. 51 S. NEWHOUSE, J. PALIS AND F. TAKENS Since the latter diffeomorphisms were obtained by modifying Anosov diffeomorphisms, he called them DA (for derived from Anosov) diffeomorphisms. Although Smale's construction was given for Anosov diffeomorphisms of the two torus, it works just as well with codimension one Anosov diffeomorphisms of the n-torus. A brief description of the construction is as follows. Let L be an Anosov diffeomorphism of T n and suppose that dim W=(x) = I for each x e T ~. Since L is conjugate to a linear toral automorphism [I6], it has a fixed point, say p. Assume that the contracting eigenvalue ofp is positive. Locally, near p, one has the usual picture of a hyperbolic saddle fixed point as in Figure 9 a. P P~ P2 \/ \ / FIo. 9 a FIO. 9 b FIO. 9 Smale proposed to modify the diffeomorphism L in a disk neighborhood N of p to obtain a diffeomorphism g with two new saddle fixed points Pl and p, on W=(p, L) and such that p is a fixed source ofg (see Fig. 93). This can be done so that g agrees with L off N, g satisfies Axiom A, and the non-wandering set of g consists of p and an (n -- I)- dimensional hyperbolic attractor containing PI and P2. In fact, as Williams pointed out in [42], one can choose g so that the foliation #'= = { W=(x, L) I x ~ T n) is g-invariant. Of course, the unstable L-foliation is no longer g-invariant. Somewhat later in a private communication with us, Williams observed that a I)A diffeomorphism g could be cons- tructed from an arc in which a saddle-node occurs. The local picture is in Figure 9 c. // // .... // t// Fio. 9 r If one chooses such an arc carefully, then one can actually make the arc mildly stable with an isolated bifurcation point. This will be proved elsewhere. These mildly 52 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS stable arcs are such that at the bifurcation point one has a saddle-node p whose stable manifold W'(p) is one dimensional and such that there is a hyperbolic periodic point q such that W~(q) c~ W~ + o. The next proposition shows that such an arc is never stable. Proposition. -- Suppose { %,}, o < ~t < I, is an arc of diffeomorphisms of T" so that ~o is Anosov with dim W~ = I for all x. Let b = b({%,}) be thefirst bifurcation point 0f{9~} and assume o < b < i. Suppose that % has a saddle-node periodic point with dim W'(p) = x and ~b has a hyperbolic periodic point q not in the orbit of p such that W"(q) n W*(p) + o. Then {~} is not stable. Proof. ~ We may assume, by perturbing {%,} if necessary, that W"(q) is transverse to W'(p). Since q can be continued to a hyperbolic periodic point q~, for ~t<b, and %, is Anosov for ~t<b, it follows that dimW"(q~) =dimM-- t. As W*(q) n W~ 4= o, this intersection is zero dimensional, and, therefore p is s-critical. Now the method of proof of Proposition (2.4) may be applied to show that {%,} is not stable. C) One-parameter families of vector fields As was pointed out in [4I], the rigidity in the conjugacy of a saddle-node arc has consequences for the stability of one-parameter families of vector fields. Consider such a C | one-parameter family{ X~,}on a 2-manifold M such that for ~ = b, X b has a saddle- node closed orbit y, i.e. a closed orbit which is attracting at one side and repelling at the other side and whose PoincarE map has first but not second order contact with the identity. We assume furthermore that "f unfolds generically; by this we mean that if S is a local cross section ofX v intersecting y, then the Poincar~ map P~ : S -~ S, ~t near b, is a saddle- node arc, see section 3 of Chapter III. Let {X~,} be another one parameter family of vector fields on M (near {X~,}) so that for ~t = b', X;, has a saddle-node closed orbit y' which unfolds generically. Let S' to be a local section of X;, intersecting y'. We assume for simplicity that neither {X~,} nor {X~} has any other saddle-node closed orbit. We say that these two families are topologically equivalent if there exist a homeomorphism h : R --~ R and a homeomorphism H~, : M -+ M, depending continuously on ~t, such that H~ maps integral curves of X~ to integral curves of X;~,). If such an equivalence (h, tt~,) exists, then h(b) = b'. Modifying H~, along the orbits of X, we may assume that H~(S) = S' for bt near b. So H~ ] S has to conjugate the saddle-node arc {P~} with {P~J. By Chapter III, section 3 this implies that Hb] S has to map X to X', whele X, X' is the unique smooth vector field on S, S' such that its time I map X1, X' l equals Pb, P;'. This means that Hbl S is determined as soon as it is determined in two orbits of Pb in S, one on each side of y c3 S. Since H b has to map separatrices, i.e stable or unstable manifolds of saddle points, to separatrices, the map Hb[ S is essentially fixed for each intersection of S with a sepa- 63 54 S. NEWHOUSE, J. PALIS AND F. TAKENS ratrix; such an intersection is an orbit of Pt. So in general {X~,} and {X~} will not be topologically equivalent if at least two separatrices approach ~" from the same side. This means that an arc { X~,} which has a saddle-node closed orbit which is approached from the same side by at least two separatrices is not stable; it has a modulus of stability in the same way as arcs of diffeomorphisms with a tangency of stable and unstable manifolds. In [7] Guckenheimer incorrectly states that some of these arcs are stable. 54 IV -- GLOBAL STABILITY 9 . Introduction In this chapter we complete the proofs of our main results concerning stability of one-parameter families (arcs) of diffeomorphisms, stated in the introduction to the paper. Necessary conditions for stability, mild and left stability at the first bifurcation point were provided in the previous chapter. We now show that these conditions are also sufficient. As a consequence we obtain a characterization for the stability of arcs containing several bifurcation values, under the basic assumption that the limit sets have finitely many orbits. The main idea here consists of a suitable construction of tubular families or folia- tions, which will be used to define topological conjugacies. We begin by describing local tubular families for hyperbolic periodic orbits. This concept will then be extended to Hopf orbits, saddle-nodes and flips. The foliations are constructed for a family { %} in ~, as defined in the introduction, especially near its first bifurcation value b. In the case of left and mild stability, this interval is of the form [~1, b], for some ~x ~ b. Other- wise, it is of the form [tzl, g~] with b E (Ezl, g~). For each Ez in such intervals, we build up tubular families or foliations requiring them to be %-invariant. We shall usually assume that the periodic orbits of % are fixed points. In fact, if a periodic orbit of % has period k, we can consider ?~ to define the foliation near one of its dements, and use %-iterates of the leaves to obtain the foliation near the others. 2. Local Tubular F~m;l;es We first recall, in a parametrized version, the notion of tubular family for a hyper- boric orbit [24], [~5]. Let ~- ~ M be a hyperbolic fixed point for r ~ ~ I. Let [~1, g~] be a neighbor- hood of ~ in I and U be a cell neighborhood of s in 1VI. If these neighborhoods are small enough, there exists a continuous mapping [[zx, ga] ~ ~ ~ x~ e U, where x~ is the unique (hyperbolic) fixed point for % in U and s x~. We denote by W"(x~,) and W'(x~,) the unstable and stable manifolds of % at x~. Given the family ( % }, V e I, we define (I) : M x I --* M by (I)(x, ~) = (%(x), tz). Definition (~,. 9 -- An unstable tubular family for { % } or 9 at (~, ~) is a continuous foliation ~" of U � [[zx, g,], such that for [z ~ [~1, ~] a) the leaves are C ~ discs raring continuously in the C k topology, I <_k< 0% and = x { } n u x { }, 65 S. NEWHOUSE, J. PALIS AND F. TAKENS 5 6 b) each leaf o~'~(y, ~) is contained in U � { ~. } for y ~ U, c) the foliation is O-invariant in the following sense O(o~"(y, p.)) D .~"(%,(y), p.) for y, ~(y) ~ U. Remark. -- For our purposes it is enough to take the leaves of the foliation to be C 2. In fact, in their global setting, we can and will construct the foliation to be C * (k > 2) when restricted to any unstable manifold of a hyperbolic periodic orbit that intersects W'(x~) x { Ix }. See Definition (3. x) of the next section, where even more differentiability is involved. The same remark applies to Definition (2.2) below. Let us see how a similar foliation can be defined at a Hopf fixed point p of %. We assume, for tx < b and tx near b, that q~, has a hyperbolic fixed point p~ near p and dim W'(p~,) = dim W'(p). As ix -+ b, ix < b, the stable and unstable manifolds of p~, in U converge to those ofp in the C k topology. So, we can use Definition (2. i) in this case, with the difference that we consider tx in some interval [~x, b]. As a natural extension of Definition (2. i ), Let nowp be a saddle-node or aflip for %. we present below the concept of strong unstable foliation for { %, } or 9 at (p, b). First we need some basic facts about center manifolds as stated in Chapter II. Let U be a small cell neighborhood ofp in M and [~q, ~ C I a small interval with b e (~q, ~). Let W ~ bea C ~center manifold for 9 at (p,b), I<k< oo. For each ~tr tJv.], W~, = W c n U x { ~ } has dimension one and W~ is C ~176 We also consider the center stable manifold W cs for 9 at (p, b), which is C k and has dimension s + 2. Here s and u are the number of eigenvalues (with multiplicity) of d%(p) with norm less and bigger than one, respectively. Both W ~ and W ~ are invariant by 0, and W~, and W~,' = W ~' t3 U � { ~t } are invariant by r Frequently, here and in the sequel, we will identify a subset V � of M � with its projection V into M. We recall that W ~ is foliated by the strong stable foliation ~'"s, with leaves s-dimensional C k discs transverse to W ~. Particular leaves are the strong stable manifolds through the fixed or periodic orbits of % near p. For each ~ e [t~l, ~], W~,' is a union of leaves of ~~'. The foliation ~-" is invariant by 9 in the following sense: if z, 0(z) z U X [~1, t~] and S is the leaf through z, then 0(S) is contained in the leaf through 0(z). Similarly, we can define the center unstable manifold W ~" for 9 at (p, b) and W ~" is foliated by the strong unstable foliation #-". Particular leaves are the strong unstable manifolds of the fixed or periodic orbits of %, near p. Our strong unstable tubular family (or foliation) is an extension of ~"" to a full neighborhood of (p, b) in M x I. For later purposes we need this extended version of ~'", but no such version of #'". Suppose (p, b) is a saddle-node. As it can be seen from Chapter II, there are two possibilities for its unfolding. In the first one, for each ~ < b and near b, there are two hyperbolic fixed points Pl,~, and P2.~, of ~, near p, which collapse into p as tt -+ b and then disappear for ~> b. We also may assume that dimW'(p,.~) = s + x and g6 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS dim W~ = s. The other possibility is similar, only the orbits Px,~ and P2,~, appear for ~ > b. Throughout this chapter we assume the first case. If (p, b) is a flip, we also assume the following of four similar possibilities for its unfolding. For each IX < b and near b, there is a hyperbolic fixed point p~, near p. For ~x ~> b, there is a hyperbolic fixed point Pl,~, and a hyperbolic period two point P2,~, near p. We assume that dim W'(pl,~) = s and dim WS(p2,~,) -= s + I. Definition (~,. ~). -- A strong unstable tubular family o~ ~' for { % } at the saddle-node or flip (p, b) is a continuous foliation of U � [~x, ~] such that a) the leaves are C k discs varying continuously in the C k topology and b) = W"(p, b) n U x { b }, b) for each ~, U � { Ix } is a union of leaves transverse to W~,', c) the foliation is ~-invariant: if o~(y, ~) is the leaf through (y, ~) eU x {~}, then r ~)) is the leaf through (?~(y), ~) ~U � These local unstable and strong unstable foliations have already been used in a similar context by several authors; see [26] for references. We provide a construction of them in Proposition (2.3) below, to give a clearer view of some of the main techniques of this chapter. Proposition (2.3). -- There exists a strong unstable foliation for {%} at (p, b), where p is a saddle-node or a flip. Similarly, there exists an unstable foliation at a hyperbolic or a Hopf fixed point. Proof. -- Let us first consider (p, b) to be a saddle-node or a flip. In U � [[xa, ~ua] let W ~' be the center-stable manifold of q) at (p, b). As before, 9 is defined by 9 (x, Ix) = (%(x), ~) and U ~p, [~xl, ~] ~b are small neighborhoods. In W ~ we take a closed fundamental domain A. This is a set with the following property. ~](A) nA=o and there is a neighborhood U' of (p,b) in W ~* such that if xeW ~~ and its negative q)-orbit leaves U', then the ~-orbit of x has an element in A, which is unique except if it belongs to 0A. Let us indicate the construction of such a fundamental domain when (p, b) is a saddle-node. Taking a C z coordinate system in the center- stable manifold of ~b at p, we can write %(x,y) = (x + x 2 -t- x(c,y) -t- r 3, [I xay 1[, [ly [t2),Ay + ~(11 xy I[, [[y][2)), where c e R 2 and []A[] < i. Here y -= o represents the one-dimensional center- x = o the s-dimensional strong stable manifold W*'(p). The stable manifold and manifold W'(p) of % at p is the half-plane x ~ o. We take in W"(p) a hemisphere H of small radius centered at p and slightly extended transversaUy across W"(p). It is easy to check from the above expression that %(H) nil---- e. For Ix1 <_ ~_< b, H~ = H X{~} also crosses W"(px, [z) and W'"(p2, Ix) transversally in U � {ix} and 8 S. NEWHOUSE, J. PALIS AND F. TAKENS ?~(H.)c~H~=o. Also, for b<lz<~, ?~(H.) c~H~-----e, where H~=Hx{Iz}. We let A~ be a closed region bounded by H., ?~(H.), and an annulus whose boundary lies in H~ c~ ?~(H~). Then take A = U A~. ~,<~<~., Ap Ao ~l (HjL) ~b (H) ~~X~-J Hy ~H Fio. Io In the case of a flip, A is diffeomorphic to A b � [[zl, ~,], where A b is an s + l-dimensional annulus. Similarly, if (p, b) is a Hopf point, A is diffeomorphic to A b � [~Zx, b], where A b is an (s -t- 2)-dlmensional annulus. If (~-, ~) is a hyperbolic fixed point, A is diffeomorphic to A~ � [~zl, g~], where A; is an s-dimensional annulus and ~ ~ (~1, ~). Let us proceed with the construction of the strong unstable foliation for a saddle-node or a flip. Over a neighborhood U1 of the exterior boundary of A we raise a fibration of class G k, the fibers being u-dimensional discs transverse to W c' and each of them contained in some U � { ~ }. Over the neighborhood O(U1) of the interior boundary of A, we have a similar fibration, the fibers being the O-images of those in U1. Restricting to smaller neighborhoods of ~A, we can extend this fibration to a full neighbor- hood of A. This is done as follows. Over a neighborhood of A, we raise another C k fibra- tion transverse to W ~' and the fibers contained in the sections U � { ~ }, but not necessarily %,-invariant. We now define a new fibration, which agrees with the first one in a neighborhood of ~A and with the second one off a slightly bigger neighborhood of ~A. Let ~1 and ~2 be the projections, into a neighborhood V of A, defined by the two fibrations. We define a C k real function p : V ~ [o, I], such that p is I near ~A and o off a small neighborhood of ~A. The required fibration is then given by the projection = Pnx + (I- P)n2. Its fibers form the leaves of our foliation. We now simply define it over W ~ -- W ~ through iterates O" or V~, all n~ o: if (y, F) -- O"(x, tz) for (x, ~) E A and the fiber through (x, ~z) is ~""(x, ~z), then the fiber through (y, ~) is O"(.~~"(x, F)) c~ U � [[zx, ~]. By the generalized k-lemma [24], [3o], ~-" extends over W ~ satisfying all the conditions of Definition (2.2). Notice that, on the center unstable manifold of 9 at (p, b), we get the usual strong unstable foliation. The cons- 58 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS truction at a hyperbolic fixed point is easier and can be done in a similar way. The same applies to a Hopf point (p, b), in which case we take the parameter ~ in some interval [~l, b]. This finishes the proof of the proposition. We point out that these unstable and strong unstable foliations are not unique. There is a degree of freedom in their construction, as shown in Proposition (2.3). This is what enables us to globalize them in a compatible way with the tubular families of other periodic orbits. Such globalizations will be performed in the next section. 3- Global Tubular FamIHes In this section we construct compatible systems of tubular families or foliations for a family {%,}, ~z near its first bifurcation point. As before, we denote by b e I the first bifurcation point, so that % is Morse-Smale for tz ~ b. From the previous chapter, {q~,} can be stable in one of our three senses only if one of the periodic orbits of ~b is an elementary bifurcation (saddle-node, flip or Hopf). Moreover, the (strong) stable and unstable manifolds of all the periodic orbits of % must have transversal intersections. We recall that, since q~ is Morse-Smale for [z ~ b, its periodic orbits can be partially ordered through the relation pi,~>pj,~, if W~(p~,~,) nW*(pj.~)oe o. See [24], [25]. We fix a total ordering for the periodic orbits compatible with this relation. If ~b has a Hopf orbit or a flip, the same ordering applies as well to its periodic orbits: there can not be any cycle, since otherwise b would not be the first bifurcation point [x8]. If 9b has a saddle-node, we may have a cycle containing this orbit. If there is only a I-cycle, the saddle-node will be counted as pj,~ and Pj+t,~, for some positive integer j. However, if there is a cycle of larger length, then the saddle-node will be counted as p~,~, Pj +j,, ~ where j and j' are positive integers and j' > I. In all cases, this ordering of the periodic orbits will be used to build up global systems of foliations for { % }, ~ in a small interval in I. This interval is of the form [~t, b] when % has a Hopf point or a saddle-node which is critical or has a cycle, and otherwise [Eli, ~] with b e (~1, ~). Let Px,~ > P,,~, > 9 9 9 ?~ Pt,~, be the periodic orbits of % for some ~ ~ [~q, g~], all of them hyperbolic except at most one which is an elementary bifurcation. For convenience, we write p~ instead of pi,~. Let Ui, U2, ..., Ut be neighborhoods of Pi,P2,...,Pt. We will consider local unstable or strong unstable foliations in U1 � [~li, ~], U2 � [~i, ~u/], ..., U t � [[ii, g~] as in Definitions (2. I) and (2.2). If the periodic point p~ is a saddle-node with no cycles or a flip, let .,~-~(pi) denote its strong unstable foliation in U~ � [Eza, ~]. Otherwise, let o~-~(p~) denote the unstable foliation in U~ � [#~, g~]. Since ~-~(p~) is a ~;invariant foliation in U~ � [~, g~], it naturally induces one in [.] ~,(U~) � {#} for [z e [#x, g~] and n e N by simply taking iterates ~, of the leaves. In the sequel, we will be using this extended (or globalized) foliation, which will be still denoted by ~z-~(pi). 60 S. NEWHOUSE, J. PALIS AND F. TAKENS Definition (3. ~ ). -- The system of foliations o~'.(pi), o~',(p,), ..., o~',(pt) is compatible when a) ira leafF of ~"(p~) intersects a leafs of ~(p~), i ~j, then FDS, b) for all i ~j, the restriction of ~"(p~) to each leaf of ~(p~) is a foliation of class C ~, ~ ~ k. Remark. -- Note that the restriction of these foliations to unstable manifolds of hyperbolic periodic orbits are of class C *. However, the unstable manifold of a non- critical saddle-node with no cycles or a flip is not a leaf of the strong unstable foliation. Thus, we do not demand differentiability of the foliations when restricted to these unstable manifolds. Let us show the existence of such systems. First we consider the case where ~ has a Hopfperiodic orbit. This is the same as the parametrized version of the hyperbolic ease. Although somewhat simpler than the saddle-node and the flip, it gives a pretty good idea of how to proceed in those cases as well. Proposition (3.2). -- If 9~ has a Hopf periodic orbit, then there exists a compatible system of unstable foliations for ~,, ~ ~ [~l, b] for some ~i near b. Proof. -- Let Pl,~ )" P,,~ > ..- > Pt,~ be an ordering of the periodic orbits of ~ for ~1 ~ ~ ~ b. We simply write p~ instead of P~.b. By induction, we may assume that a compatible system of foliations ~-"(Px), ..., ~-~(P.) has been constructed in neighborhoods U l� ...,U.� [~1, b], p~Ui for I <i<n. Let us build ~"(P,+ 1)- We will adapt the proof of Proposition (2.3) to guarantee the compatibility condition. Near P.+,, we consider a closed annulus A b as a fundamental domain for W'(p,+,) and denote by Oe.A b its exterior boundary. Then, the annulus A~, with boundaries 0exAb � {~} and ~,(O,.A b � {~}) is also a fundamental domain for W~(p.+a, ~), p~ e [~l, b] (we take ~1 closer to b if necessary. If W~(p,) intersects W'(p,+l) it must do so transversally. Thus, we can take a C u fibration of W"(p,) near the exterior boundary of Ab, whose fibers are discs transverse to W~ with the same dimension as W~(p,+,). The image by ~b of this fibration induces a similar one near the interior boundary of A b. On the other hand, it is easy to get a second C ~ fibration of W"(p,) in a neighborhood of all of its intersection with Ab, if we do not require it to be %-invariant. However, as in Proposition (2.3), this second fibration can be deformed to agree with the first one near the boundaries of A n and, thus, it becomes ~b-invariant. We now want to fiber in a similar way the leaves of o~"(p.) near A~,, ~1 ~ ~ ~ b with ~x close to b. First, we observe that, in U, � [~l, b], the leaves of ,~"(p,) are C k imbed- dings of the disc W~(p.) n U,, continuously parametrized by (W'(p,) n U,) � [~1, b]. Thus, by restricting this parameter space, we have that the leaves of ~'"(p,) near A~ are (I k close to W"(p,). Using this parametrization and the fact that the foliation o~'~(p,) is ~-invariant, we can fiber as above the leaves near W*(p,) n A b. We get a ~-invariant fibration with C k fibers transverse to We(p,+l, ~), ~i ~ ~ ~ b, and varying conti- nuously in the fil k topology. By construction, each fiber is contained in some leaf of 60 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6i o*'"(p,). Also, the fibration is C k when restricted to each leaf of o*'"(p,). Now we take #-"(~,-1). There are two cases to consider. If W"(p,_l) intersects W~ but not W'(p,), then W"(p,_ 1) n A b is compact and disjoint from W"(p,) n A b. So, we can proceed as before, fibering the leaves of ~-"(P,-a) near W~(p,_l) n A. Let us now suppose that W"(p,,_ 1) intersects W*(p,). Since o*-"(p,_ 1) and o*'*(p,) are compatible by the induction hypothesis, the fibration of o*"(p,) in a neighborhood V of W"(p,) n A b is also a fibration of o~-"(p,_ 1) in V. On the other hand, B = W"(p,_ 1) n (A b -- V) is compact. So, o*"(p,_a) can be fibered as above in a neighborhood of B. As in Proposition (2.3), we can average these two fibrations near 0B to get a desired one in some neighborhood of W"(p,)u W"(p,_l) intersected with A b. Its fibers form the leaves of our foliation. We repeat the argument to all o*'~(p~), t < i < n -- 2. Once we have the q~-invariant and compatible foliation o*-"(p,+a) near A b � [~1, b], some txx close to b, we just consider its positive iterates by % (or r By the generalized X-lemma [24], [3o], it extends to the foliation W"(p,+ 1.~,), ix 9 [ix1, b], of the center unstable manifold of q> at (P,+I, b). This finishes the construction of the folia- tion #'~(p,+ 1). The proof of the theorem is complete. Let us now consider the case where % has a saddle-node. Let Pt,~, > P2,~ > ... > Pt,~ be an ordering for the periodic points of q~, bit <__ ~ < b. We first consider the case where the saddle node is non-critical and % has no cycles. We assume that pj,~, and Pj+a,~ coalesce at ~ = b, giving rise to the saddle-node pj = Pj+t. We shall construct a compatible system of foliations for {%,}, with ~ in some interval [~t, V,.] and b 9 (~t, t~). The foliation at the saddle-node pj =Pj+t will be a strong unstable foliation ~(pj). Actually, the construction we just performed in Proposition (3.2) can be adapted to the present case as well as to the flip bifurcation. However, to prove the stability of the family, we also construct a one-dimensional center foliation in a neighborhood of the saddle-node pj = pj+ t in the center stable manifold of O at (p~,b). As usual, O:MxI--~M is defined by r ix) = (q~(x),lx). This center foliation o~" should be q~-invariant and compatible with the unstable folia- tions o~(p~), i<j, in the sense of Definition (3. I). Proposition (3- 3). -- If % has a non critical saddle-node p~ = pj + 1 and no cycles, then (a) there is a compatible center foliation ~'~ defined in a neighborhood of the center stable manifold W ~" of * at (Pj.b), (b) there is a compatible system of unstable foliations ~"(pi) for 9 or ~, ix 9 It� ~] for some ix1 < b and ~ > b. The foliation we consider at pj = pj + t is a strong unstable one, o~"", (c) the union of leaves of ~"" through a leaf of o*" forms a C t submanifold. Proof. -- Since Pl > P2 > 9 -- > Pj- 1 are all hyperbolic, Proposition (3.2) provides a compatible system of unstable foliations o*'~(px), ~-"(p~), ..., o~"(pj_l). The leaves of these foliations are taken to be C1 k, k > 2. Let us construct a center foliation ~'* 61 62 S. NEWHOUSE, J. PALIS AND F. TAKENS at the saddle-node; its leaves will be C ~'-~. In the center stable manifold of r at (pp b), we consider a fundamental domain as in Proposition (2.3). We define the center foliation ~'~ as the integral curves of a vector field X satisfying the following properties. X is tangent to the leaves of ~'*(pi) and it is C k- 1 along these leaves for all i < j such that W*(pi) intersects W'(pj). The vector field X is also transverse to the strong stable foliation of~ at p~ and d~-invariant. To get such a vector field, we proceed by induction taking i =j -- I, j -- 2, ..., I. If W~(p~_l) intersects W e', we take a C k-a vector field X tangent to W"(pj_l) near its intersection with the exterior boundary of the fundamental domain A. Near the interior boundary, we just consider d~(X). It is easy to extend this vector field X to a full neighborhood of W~(p~_l) t3 A in W~(p~_l). We now want to define X along the nearby leaves of ~-"(P~-I). As we noticed in the proof of Proposition (3. ~), these leaves are C k imbeddings of a disc in W"(pi_l), conti- nuously parametrized by their intersection with W'(pj_x). So we can project X into the leaves of ~"(Pj-1) near the exterior boundary of A, consider its image by d~ and extend it across A as above. Due to the fact that the saddle-node is noncritical, X is transverse to the strong stable foliation of 9 in the center stable manifold. By induction, let us suppose X defined along the leaves of~'~(pi + 1), 9 .., ~-~(Pj- 1) near the fundamental domain A. If W"(p~)r3 A-----r we proceed to the next foliation ~"(P~-I). If W~(pl) r3A. o, but W"(pi) nW'(pk) = o for i<k<j, we can proceed as before since WU(p~) r3 A is compact and disjoint from W~(pk) t3 A. Finally, let W~'(p~) n W'(p,) . o for some i < n < j. The vector field X is already defined in a neighborhood V of W"(pk) in A for i < k < j. Since .~'~'(p~) is compatible with ~'~(Pk), X is tangent to the leaves of o~-~(pi) in V. But W~(p~) n (A -- V) is compact, so we are again reduced to the previous case. Thus we can construct X as desired in a neighbor- hood of A in the center stable manifold. Now we take the integral curves of X and their positive iterates by r By the ),-lemma [24], [3o], this partial foliation extends to center manifolds and together they form a center foliation as asserted in (a). Parts (b) and (c) follow as in Proposition (3. ~) for the hyperbolic periodic points and also as in Proposi- tion (~. 3) for the saddle-node. In this last case, we raise the strong unstable foliation from the center foliation we have just constructed. The proof of Proposition (3.3) is complete. Remark. -- The center foliation ~~ is not a " classical " foliation in two ways (1) as usual in the present work, we only required that the leaves of ,~-~, which are G ~- 1 with k > 2, should vary continuously in the C ~- 1 topology. We did not demand the projection along the leaves into a transverse section (like a leaf of the strong stable foliation) to be differentiable, (2) ,~'~ has singularities, as shown in the picture FIo, xl 62 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 3 Now let us consider the situation of a saddle-node p~ = p~ +j, for Vb which is critical or lies in a cycle and j' > I. In the case the saddle-node is bicritical or it is part of a cycle, we wish to prove the stability of the family { q~ } for V~ _< b (left stability). When the saddle-node is critical but not bicritical and there is no cycle, we wish to prove that { q~} is midly stable for [z < ~1, some ~1 > b and near b. Again, since %, is Morse- Smalefor b< tz< Vtl, some [zl>b , { r } is stable in this range of the parameter. This follows from [25] or the next section. Thus, it is certainly enough to show that {?~} is left stable at b. In conclusion, we can treat these three cases in the same way. As before, we need to construct a compatible system of unstable foliations for { q~}, F <__ b. Proposition (3.4). ~ If % has a saddle-node pj =pj+j,, j' > z, which is critical or, has a cycle, then there exists a compatible system of unstable foliations for V. <_ b. The leaves of ~"(Pi+J',~,) are one dimension lower than those of pj,~. Proof. ~ Similar to that of Proposition (3.2), using the unfolding of the saddle-node as in Proposition (2.3). In this case the leaves of ~"(p~, t~) cover a neighborhood of W~'(p~, ~) --{pj,~,} near pj,~ where W*'(pj,~) is the strong stable manifold of pj,~. The leaves of ~-"(p~,~) intersected with the center stable manifold W~,' are repre- sented by horizontal lines in the pictures below. The intersections of the leaves of ~"(pj+j,,~) with W~ ~ are represented by points. "x,,x" \~'K f,- / I / / i It I , /\ /\ VN W cs 3,, ~ <b vn w? Fio. x2 Notice that the foliations ~z-,,(pj,~) and ~"(pj+j,,~) can be constructed to be compatible in a neighborhood V of WS~(p~) --p~ in M � [~1, b] for some [zl< b. Proposition (3- 5). -- If @b has a flip point pj,, then (a) there is a compatible center foliation ~ defined in a neighborhood of the center stable manifold of rb at (p~, b); 63 S. NEWHOUSE, J. PALIS AND F, TAKENS (b) there is a compatible system of unstable foliations ~"(pi) for ~, ~ ~ [~l, ~] for some ~l < b and g~ 2> b; the foliation we consider at the flip pj is a strong unstable one, ~; (c) the leaves of ~"' through a leaf of ~" form a C 1 submanifold. Proof. -- The proof is entirely similar to that of Proposition (3.3)- We finish this section by summing up in the next theorem the results we have obtained on compatible systems of unstable (strong unstable) foliations. Theorem (3.6). -- Let b be the bifurcation point of the family {%} in ~. Let @ : M � I -,'. M be defined by r ~) = (%,(x), ~). Then, (I) if% has either a noncritical saddle-node and no cycles or a flip, there exists a compatible system of unstable foliations for r or {%}, ~ < [z i for some ~l near b and ~l 2> b. The unstable foliation at the saddle-node or the flip is a strong unstable one, (2) if % has a saddle-node or a Hopf periodic orbit, there exists a compatible system of unstable foliationsfor @ or {9~}, ~ <__ b. 4- Stability We culminate this chapter by showing the stability of the families of diffeomor- phisms {%,} in ~. Such families were introduced in Chapter III, studied there and in the previous sections of the present chapter. A family { %} in M is stable, mildly or left stable according to the structure of the diffeomorphism %, where b is the first bifhr- cation point of {~}. More specifically, the stability depends on the structure of the non-hyperbolic orbit of %. We stress again that the transversality of the stable and unstable manifolds of the periodic orbits is necessary even for left stability, as was proved in Chapter III. Our results on stability will follow rather naturally and in a unified way from the existence of compatible systems of (strong) unstable tubular fatuities or foliations. Fitting together these foliations provides a kind of global coordinate system from which the topological conjugacies can be constructed. We define the conjugacies inductively on the stable manifolds of the periodic orbits, these orbits being totally ordered as in section 3. We will make use of the following two known results. The first one states that is open in the set ~ of C ~~ one-parameter families of C ~ diffeomorphisms of M with the C ~ topology. It corresponds to Theorem (2.5) of [i9] and Theorem (3. i) of [20]. The second result is an easy extension (a parametrized version) of the Isotopy Extension Theorem [~3]. Theorem (4. x). ~ Let {%} e ~ have the first bifurcation point b. There exists a neighborhood U of {~?~} such that if {~} ~ U has first bifurcation point ~, then {~,} e ~ and % and "~ have the same elementary bifurcation. Moreover, there is an order preserving one to one correspondence between the periodic orbits of % and "~. fi4 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 5 Let N be a C' compact manifold, r ~ x, and A an open subset of R ~. Let M be a C | manifold with dimM~dimN. We indicate by C](NxA, M� the set of C * mappings f: N � A -+ M � A such that n ---- ~'f, endowed with the C * topo- logy, ~ <k<r. Here, n and 7:' denote the natural projections n:NxA-+A, ~':M� Let Diff~(MxA) be the set of C *diffeomorphisms @ of MxA such that ~' -~ n'@, again with the C ~ topology. Theorem (4.~,). --Let i e C](N � A, M � A) be an imbedding and A' a compact subset of A. Givenneighborhoods U of i(N � A) in M � A andV of theidentity in Diff~(M � A), there exists a neighborhood W of i in C](N x A, M � A) such that for each j ~ W there exists , e V satisfying ~i =j restricted to N � A' and ?(x) -= x for all x r U. Now we prove the main theorem of the present chapter. Theorem ('t. 3). -- Let { ?~} ~ ~ with first bifurcation point b. Then I) if % has a flip or a noncritical saddle-node with no cycles, {~} is stable, 2) if ?b has a saddle-node which is not bicritical and has no cycles, {q~} is mildly stable, 3) if ?~ has a Hopf periodic orbit or a saddle-node which is bicritical or has a cycle then {@~} is left stable. Proof. -- First, we observe that the statement in part (2) can be proved as in part (3). In fact, in case (2) we have that {q~} is Morse-Smale for b < ~ ~ ~tt, some [z t > b, and so it is stable in this range of the parameter. This last fact is an easy consequence of the proof that 1Vforse-Smale diffeomorphisms are stable. By the same reason, in all cases it is enough to consider the stability of{ ~%} in ~ for [z near the first bifurcation point. Let Pl,~, ~ Pz,~ ~ ... ~ Pt,~ be a total ordering of the periodic orbits of ?~ for < b. As in section 3, this ordering can be naturally extended for [z = b and even for [z > b when 9b is as in parts (t) and (2) of the statement. By Theorem (3.6), we can take a compatible system of (strong) unstable foliations ~'u(pl) , ~u(pz), ..., ~(Pt) defined in 1V[ � [~t, ~z~] or 1V[ � [~zl, b], for some ~1 < b and g~ ~ b according to case (I) or (3). In case (I), the foliation is a strong unstable one at the flip or the saddle- node. Recall that this strong unstable foliation is constructed from a center foliation .,~-~ in the center stable manifold of@ at theflip or the saddle-node, where 9 : IV[ � I -+ M � I is defined by r ~) = (?~(x), ~z). By Theorem (4. I), for a nearby family {~'~} with first bifurcation point b, we can consider a corresponding compatible system of (strong) unstable foliations and a compatible center foliation in case (i). Also, given a repara- metrization (a homeomorphism) p : I ~ I near the identity and p(b) = ~, there is an order preserving continuous correspondence between the periodic orbits p~,~ and ~.p(~,) of q~, and ~p(~,), for ~ ~ [V~, ~] or V~ ~ [~z~, b]. This defines the conjugacy on the periodic orbits. In the construction of a global conjugacy we can take this repara- metrization quite arbitrarily, except in the case of a noncritical saddle-node with no 9 66 S. NEWHOUSE, J. PALIS AND F. TAKENS cycles. In this situation we must choose p as in Chapter II, so that we have a (continuous) conjugacy between (I) and @ restricted to center manifolds at the saddle-nodes. Our global conjugacy H will be constructed on M x [~l, ~] or iV[ � [~ti, b] inductively on the stable manifolds of the periodic orbits and, in case (i), at the center stable manifold of (I) at the saddle-node or the flip. Since these stable manifolds cover all of iV[x{~t} for each ~z~[~i,g~] or Ez~[~l,b], H will be defined on all of MX [~zx,~] or Mx [~t i,b] and maps onto 1Vfx [P(~i),P(~)] or MX [p(Ezx),b']. We fix fundamental domains A~ for W"(p~) or WCS(pk) with exterior boundaries transverse to all WU(pi), i < k ~ t. The conjugacy H will be constructed with the following properties: a) H(Mx{A) =Mx{p(~)}, ~in MX [~1,~] or in MX [~,b], b) it sends leaves of ~-"(pi) into leaves of ~r,(~), I < i < t, c) it is differentiable along each leaf of o~-"(pl) off the stable manifold of (I) atpl, I < i < g, d) if W"(pl) n W*(pk) ~e o, i < k, then for each leaf F of ~-"(pl), in a neighborhood of F n Ak, H is C a near the inclusion map. As the first step H takes the sources of{q~} onto the sources of {~'~(~1}. Suppose H has been constructed on the stable manifolds of @ at Pi > P~ > -.- > P,. Notice that the space of leaves of ~"(pi) is parametrized by the intersection of the leaves with W'(pi) or W~(p~). Thus, in particular, H defines a map from the space of leaves of ~'~(pl) onto the space of leaves of ~"(~) for I < i < n: F ~ ~"(p,) is associated to F ~ o*-"(~) if H(F n W*(p~)) F c~ WS(~) Let us now consider W~(P.+i), W (P.+z) ifP.+i is either hyperbolic, a Hopf orbit or a saddle-node which is critical or has a cycle. Let Ab-~A.+in(Mx{b}) and Ag=A.+1n(Mx{~)). If W"(p.) nAb4=O, we choose H as a diffeomorphism near the inclusion of a neighborhood of Wt'(pn) n 0e.Ab in W~(p.) onto a neighborhood of W~(~.) n 0o.Ag in WU(ff.). Such a ditTeomorphism exists because W"(p.) n A b is close to W"(~) n A~ for {9~} near {~,}. Near the interior boundary of Ab, H is defined by H% = ~g H. From Theorem (4. ~) this partial diffeomorphism can be extended to all of WU(p.) n A b. To extend H to the leaves of ~'u(p.) near A~+ l, ~ <_ ~ ~ b, we proceed in an entirely similar way using Theo- rem (4. ~), since we already know which leafof~(~) is associated to a given leaf of #-'*(p,,). This completes the construction of the conjugacy on the intersection of the leaves of ~'u(p.) with the fundamental domain A.+ 1. Next, we take ~(Pn-1) and suppose that W'*(p~_l) n A b 4: o. There are two cases to consider. If WU(p,,_ i) c~ W'(pn) = o, then W~(p._ i) n A b is compact and disjoint from WU(p~) ~ A b. Thus, the cons- truction of the conjugacy on the leaves of .~-u(P.-1) restricted to A.+a is the same as on the leaves of ~"(p.). So, assume that W~(p._ i) n W'(p.) 4: o. Since ~-"(p,~_ i) and 9 ~'u(Pn) are compatible, H is already defined on the leaves of~'~(i0n_ x) in a neighborhood V of W"(p.) c~A n inA.+i. But B=W"(p._l) n(A b-V) is compact. So, we can extend H to B using Theorem (4.2) and also to the leaves of ~*"(p._ i) in a neighborhood of BinA.+l. We repeat the argument to all ~(p~), I <i<n--2. IfP.+iis a 66 BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 7 Hopf orbit or a critical saddle-node with no cycle, then H is defined on A.+ i satisfying H?~, = ~p(~,)H or H 9 = ~H, ~1 ~ ~ ~ b or ~1 ~ ~ ~ ~ with ~1 < b and b < ~,. The same equation allows us to define H on all of the stable manifold of @ at p.+i. If the saddle-node occurs in a cycle so that P.+x = P.+ l+j' for some j'~ I, then we have only defined H on a neighborhood V of W"(p.+l,b)--{P.+a,b} in W~(p,,+l). We continue as before forp.+l+~with j <j'. Whenwecome top.+l+~, , the conjugacy H as already been defined on the part V of W'(p. + 1+4'). We continue the process as above obtaining a conjugacy Ha defined on (k<.~l+j' W"(Pk))n W'(p.+l+~, ) such that H i ~-H off a neighborhood of W'(p.+l+i, ) --V in W'(P.+l+j,). Thus, H can be defined on all of WS(p.+l). Now we consider the case where P.+l is a flip or a non-critical saddle-node with no cycles. Here, H will be defined in the center stable manifold of @ at p.+ 1 as the "product " of two partial conjugacies. One of them, which we call H e, is defined at the center manifold of @ at p. + x, as constructed in section 3 of Chapter II. It corresponds to a conjugacy on the space of leaves of the strong stable foliations ~"(p.+ 1), ~r.+ aJ. The other, which we call H', is to be defined on the space of leaves of the center folia- tions ~"(P.+i), ~'~(~+ 1). However, since the center foliations may have several leaves going through the periodic orbits P.+I, ~+i, we cannot express the stable manifolds of such orbits exactly as the product of these two foliations. That is the reason we will make a slight modification of the center foliation ~-'(P.+I) already constructed. The second partial conjugacy H' will then send leaves of ~-"(p.+ i) onto leaves of the modified center foliation for ~. + 1. To do this, we first consider a continuous family of leaves { F~, }, _ _ ~r.+ l, ~), such that F~, meets all the leaves of the center foliation ~'~(P.+i, ~). We also choose a family of disks {D~}, D~,CF~,, such that the exterior boundary 0~. D~, of D~ is transverse to the unstable manifolds W"(p~, ~) for all I <i<n and each ~e[~l,~]. We now form a fundamental domain A= UA~ for the center stable manifold of/,. + 1 as follows. For each ~, the exterior boundary 0~. A~ is made of the disk D~, and a " cylinder " formed with the leaves of o~-"(p.+ i, ~) through the points of 0e. D~. Notice that H" determines a corresponding family { F~(~)} of leaves of ~z-~t~'~r.+ lJ~. We then take a family of disks {Dp(~,)}, D~(~,)C F~(~,) for each ~, and construct a similar fundamental domain A = U .~(~) for the center stable manifold of~+a. We now define H" : U D~ -+ U Do(~,) as follows. Of course, we want it to be compatible with the unstable foliations ~-"(pl) and the conjugacies already defined on the stable manifolds W'(pi), I < i < n. Near the exterior boundaries of D~, D~(~) we proceed by induction on the indices i = n, n -- z, ..., ~. We start with a diffeo- morphism H":V ~V near the inclusion map, where V is a neighborhood of W'*(p~) ~ 0exD ~ in W"(pn) c~ F~ and V is a neighborhood of W"(~) ~ 0exD0(~) in W"(~.) c~ F0(~)" Similarly for the intersections of the leaves of ~"(p.) with 0o.D~ which are mapped onto the intersections of the corresponding leaves of o~-"(~.) with 0,. D0(~)" Notice that the correspondence between the leaves of ~"(p.) and those of ~'~(~.) is 67 68 S. NEWHOUSE, J. PALIS AND F. TAKENS determined by H:WS(p,) -+ Ws(p,) already defined. The extension to the leaves of ~"(P,-1), 9 9 -, ~"(Pl) is done in the same way as in the previous case. In order to define H ~: U D~ --> U D~(~), we consider the annuli A~CD~ and ~(~)CDp(~), each ~z E [Or1, W], obtained as follows. The exterior boundary of A~, is 0e.D ~ and its interior boundary is the projection of ?~,(Oe, D~,) into D~ via the leaves of ~-c(p,+l, V~). Similarly for "~C~)" Notice that H~:Oe~A~,-~ 00.Ao/~) constructed above induces a map H* : 0i.A ~ -+ Oi, A~(~). In fact, for x ~ 0i.A ~ let g, be the leaf of ~'*(P,+I, ~) through x and let y be the intersection of ~-t(t,) with 0e.A~. If 7 is the leaf of ~-"(~+ a, p(~z)) through H*(y), we set H"(x) = z, where z is the intersection of ~p(~)(7) with Oi, A~c~). We now extend H' first to all of A; using the Isotopy Extension Theorem as before. Finally, we can extend H 8 to all of D~ sending leaves of #z-,,(p~, ~z) to leaves of ~-"(~, p(tz)), i < i < n. Again, this can be done as before since W"(p~, t~) and W (p~, O(~)) are transverse to D~ and D0(~) , respectively. At this point, we would like to define the conjugacy H : W**(p,+l) ~ W"*(p,+I) as the " product" of H* and H' using the strong stable and the center foliations. To do this we have to modify the center foliation ~'*(P,+I) in (P,+x). For each V~ ~ [tzl, Vt,], it is enough to do so in the region bounded by Dp(~)- ~l~), ~(~)(Dp(~,)) and the cylinder formed with the leaves of ~*(~+~) through points of 0i,A~(~, ). Let ~q :Do(~)- be the homeomorphism defined by ~q(x) =y, where (H*)-~(x) and q~(H')-t~]-~](y) belong to the same leaf of ~*(p,+x, ~). Let ?,: D0(~)- A~(~)~(~,)(D0(~)) be defined by ~,(x) = z, where x and z belong to the same leafofo~-*(~+x, p(~z)). If 2, = ~ then no modification of ~-*(~+~, O(~)) is needed. So let us change o~'0(~,+~, p(iz)) to get the second map to be equal to the first. Notice that ~q(x) and ;~(x) belong to the same leaf of ~-"(~, p(~)) for some ~ < i < n. Moreover, ~,~-~ along such a leaf is C t near the identity map. Let X be the vector field whose integral curves are the leaves of t~a o*-*(~+~, p(tz)). We modify X near but off q~o(~)(D~(~)) so that the corresponding map ), satisfies 7,~- ~ --~ I on @"~(~)(Do(~)). Observe that the modification required for X along W (p,, p(~z)) is well known and it can be performed in a parametrized way along the leaves of ~"(~,, p(~)). Using the fact that the foliations ~"(~, 9(~)) are compatible, we proceed by downward induction on the indices I < i < n. It is clear that the new center foliation coincides with the previous one on the boundary OA = O OAo(~, I of the fundamental domain A. Once it is defined on A, it can be defined on all o ,, ~/,, + x) simply through iterations by ~, where ~(x, tz) = (~"~(x), ~) as before. Now we can W e*t~" '~ using the center foliation, the strong stable define H from W=(p,~+~) onto ~v,+~ foliation and the conjugacies H" and H ~ on their spaces of leaves. H is clearly one to one, onto and also continuous since these foliations and the maps H 8, H ~ are continuous. Moreover, H is differentiable along the leaves of #-"(pl), I < i < n, in W**(p,+l). In fact, ~c was constructed in section 3 to be differentiable along these leaves; the same is true with the modified #z-* and the map H ~ constructed above. Being codimension one, the foliation 9~'** is C t (see section ~, Chapter II). Finally, the differentiability A~(~,)-*~"~(~,)(D~(~,)) BIFURCATIONS AND STABILITY OF FAMILIES OF DIFFEOMORPHISMS 6 9 of H a off the periodic points is in section 3 and 4 of Chapter II. Together these facts imply our statement. Thus, the proof of the induction step is finished. Our map H is defined on M � [~1, ~] or M � [~1, b] onto M � [P(~I), P(~,)] or M � [P(~I), ~], it is clearly one to one and satisfies the conjugacy equation H~ = ~'o(~)H. Let us prove that H is continuous. We have to show that H is continuous at the stable (center stable) manifolds of the periodic orbits of {q~}. Here we indicate by W 8 both the stable and center stable manifolds; also, let 11 indicate either [~1, ~] or [~1, b]. By construction, H is continuous along these manifolds and, in particular, along the stable manifolds of the sinks. By induction we may assume that H 9 W ~ W is a homeomorphism, where W and W are the union of Wa(pk,~,) and Wa(pk,~(~,)) for i< k <_/ and ~ EI 1. Let us now show that H is also continuous at Wa(p~,~,). Consider a sequence (x,, ~,) ~ (x, ~) ~ Wa(p~,~), x, ~ M and ~, E 11. Since H restricted to Wa(p~,~,) is continuous, we may assume, via a subsequence, that (x,, ~,) ~ W for all n. Let F,, F be the leaves of~"(p~) containing (x,, ~,), (x, ~) and F,, F the leaves of ~"(~) contai- ning H(x,, ~,) and H(x, ~), respectively. We have that F, -+ F and since H is conti- nuous restricted to the stable manifolds, we have F, -+ F. Thus, it is enough to show that the sequence H(x., ~,) accumulates on W (p~,~(~)). In fact, H(x,, p~,,) cannot accumulate on W because H : W -+ W is a homeomorphism. Also, H(x,, ~,) cannot accumulate on the union Z of W (p~,~(~)), I <__j < i, because F, and Z are far apart. This proves our assertion and so H is continuous on all of M � I~. This finishes the proof of the theorem. We now complete the proof of our second main theorem stated in the introduction, the first part of which was done in section ~ of Chapter III. Let us denote by ~ C ~ the set of arcs { ~, } such that the limit set of each ~, has finitely many orbits, ~ ~ I = [0, 1]. We also denote by 5~CCg the set of arcs {~?~,} such that there are only finitely many bifurcation values for { @~,} say bx, ..., b, in (o, I) and for each ~ < i < s, %; has the following properties: -- all stable, strong stable, unstable and strong unstable manifolds intersect transversally -- %i has no cycles and has exactly one non-hyperbolic periodic orbit, which is either a flip or a non-critical saddle-node; this non-hyperbolic orbit unfolds generically. As we mentioned before, it turns out that for arcs { ~ } ~ 5r ~ is a Morse-Smale diffeomorphism if ~ is not a bifurcation value. Theorem (4.4). -- The arcs in 5P are stable. Proof. -- As in Theorem (4- i), it follows from [i9] , [2o] that 5P is an open subset of~. Thus, if {~} ~ 5r and has bifurcation values bl, ..., b a in (o, I), then a nearby t t t arc { @~,} also belongs to 5r and has nearby bifurcation values bl, ..., b, in (o, i). More- over, the non-hyperbolic periodic orbits of %1 and ~'b} are both saddle-nodes or flips. To produce a conjugacy between {q~,} ~ 5r and a nearby arc {~%}, we first assume that 69 7 ~ s. NEWHOUSE, J. PALIS AND F. TAKENS 9~ = r for ~ ~ I -- U, where U is a small open subinterval of I containing at most one bifurcation value. In this case, by Theorem (3-6) we can construct unstable tubular families or foliations for { r } and { q~ } and ~ ~ V, where V is art open subinterval of I containing the closure U of U. Moreover, using the Isotopy Extension Theorem (4-2), we may construct these unstable foliations to be the same for { r } and { q~'~, } if ~ e V -- W, where W is an open subinterval of I such that U C W and W C V. Again using the Isotopy Extension Theorem, the construction of the conjugacy { h~, } between { q~ } and { q~'~, } for ~ ~V, as performed in Theorem (4.3), can be done so that h~, is the identity map on IV[ for ~ ~ V -- W. Since r = r for ~ e I -- U, we can extend this conjugacy to all of I by defining it to be the identity for ~ ~ I -- V. Finally, let {Ui}, I ~ i < n, be a covering of I by small subintervals, each containing at most one bifurcation value. It is immediate that we can decompose any small perturbation of{ q~,} into perturbations, each with support in one of the subintervals U~. So the construction of the conjugacy between { r and a nearby arc is reduced to the previous case. The proof of theorem is complete. REFERENCES [I] A. A. A~mRoNov, E. A. L~.o~rrovlc~, I. I. Go,oN, A. G. MAmR, Theory of bifurcations of dynamic systems on a plane, Isr. Program Scientific Transl., Jerusalem, x97x. [2] L. BLOCK, J. FV.ANmZ, Existence of periodic points of maps of S I, Invent. Math., 22 (1973), 69-73. [3] Th. BR6cmm, L. C. LANDER, Differentiable germs and catastrophes, London Math. Soe. Lecture Notes, 17, Cambridge Univ. Press, 1975. [4] P. BaUNOVSKt, On one-parameter families of diffeomorphisms, I and II, Comment. Mat. Univ. Carolinae, 11 (3) (x97o), 559-582; 12 (4) (x97t), 765"784 9 [5] P. 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I.M.P.A., Rua L. de CamiSes 68, Rio de Janeiro, Brazil. Department of Mathematics, University of Groningen, Holland. Manuscrit re~u le 22 avril i98t.

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