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AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS by DAVID RUELLE Abstract. -- Analytic functions are introduced, which are analogous to the Fredholm determinant, but may have only finite radius of convergence. These functions are associated with operators of the form ]* ~z(dc0) -~co, where .~'~(x) = ~po~(x).~(~bo~x), ~ belongs to a space of H61der or C r functions, ~o~ is H61der or C r, and ~co is a contraction or a C r contraction. The results obtained extend earlier results by Haydn, Pollicott, Tangerman and the author on zeta functions of expanding maps. 1. Assumptions and statement of results The theory of Fredholm determinants (see for instance [1O]) has been extended by Grothendieck [5] and applies to linear operators ~ in certain suitable classes. One associates with ~U an entire analytic function d~, called the Fred.holm determinant, such that (1 -- zz,',~) -t = .iV'(z)ldx(z ) where.A/" is an entire analytic operator-valued function. In what follows we shall obtain results of the same type. The radius of convergence of the " determinant " will possibly be finite rather than infinite, but larger than the inverse of the spectral radius of ~r The type of extension that we shall obtain concerns operators ~ with a kernel K(x,y) which is allowed to have ~-singularities of the type q~(x) 8(y -- +(x)), where and ~ have certain smoothness properties and ~ is a contraction. Operators of this sort arise in the theory of an expanding map f (or more generally of hyperbolic dynamical systems), and the Fredholm determinants ar~ then related (as we shall see) to dynamical zeta functions which count the periodic points off, with certain weights. It is desirable to understand the analytic properties of the zeta function and Fredholm determinants because they are closely related to the ergodic properties of the dynamical system defined by f (see [13]). The hyperbolic case of contracting or expanding maps considered here is that for which the most detailed results are known, but extensions to non_hyperbolic situations are possible, as the work of Baladi and Keller [1] on one-dimensional systems indicates. 176 DAVID RUELLE Let ~> 0, 0< 0 < 1, and let X be a compact metric space. We denote by C ~ = C~(X) the Banach space of (uniformly) ~-H61der functions X ---> C with the usual norm. We assume that V C X, ~b : V ---> X and q~ ~ C ~ are given such that + is a contraction: d( x, <<. o d(x,y) and ~ has its support in V. A bounded linear operator 5a on C" is then defined by l q~(x).O(+x) if x EV, (oLd~ (x) = 0 ifxCV. The operators ~ which will interest us are integrals of operators of the form .LP: (1.1) ~ = f~(do~) *~q'o where .oqr is defined with V,~, qb,o, q~,o as above, and where ~z is a finite positive measure (which we may take to be a probability measure). The following will be standing assumptions: (i) f~(d~o) II ~o II < ~ where II II is the norm in C'; (ii)Tkere is $ > 0 such that, for all co, Vo, contains the 3-neighborhood of the sup- port of q~o,; (iii) co ~--~V,~, +o, ~0,o are measurable. (Using (ii), and possibly changing 3, we may assume that there are only finitely many different V~'s, and that they are compact subsets of X. We may take as measurability condition the assumption that o~ ~ V,o , (~, x) v.-+ +,o(x), q~,~(x) are Borel functions.) We write (1.2) ~,, = fEz(de01) ... or(d%,) ~O,m(X(~)) ~,~m_,(+,omX(~O)) ... ~'~,(+'~2"'" +'~ x(G))' where the integral extends to values of ox, ..., o,, such that +,~ +~... +,o,, has a fixed point, which is then necessarily unique, and which we denote by x(G). A zeta function is then defined through the following formal power series ~o Zm (1.3) ~(z) =exp Z --~,,. m=l m 1.1. Theorem. -- Let t a~g'l denote the operator obtained when ~,~ is replaced by ] q~,~ t in the definition of X', and let e ~ be the spectral radius* of [.9U [. The spectral radius of ~U is then ~< e P, and the part of the spectrum of X" contained in { X : ] X ] > 0 ~ e e ) consists of isolated eigen- values of finite multiplicities. Furthermore, 1/~(z) converges in (1.4) {z:lz[O~eV< 1} * The proof (section 2.5) shows that e P is also the spectral radius of [ ,xr" [ taken with respect to the" uniform " norm I I I I 9 AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS 177 and its zeros in this domain are precisely the inverses of the eigenvalues of ~, with the same multi- plidties. We may thus write (1 - = where .A/" is a holomorphic operator-valued function in (1.4). The proof of this theorem is given in Section 2. 1.2. Remarks. -- a) We see that 1/~(z) plays the role of a Fredholm determinant. However, ~(z) depends on the decomposition (1.1) and not just on the operator ~. We shall obtain a " true " determinant in the differentiable case below. b) Let E be a finite-dimensional e-H61der vector bundle over X (i.e., E is trivia- lized by a finite atlas, and the transition between charts uses matrix-valued e-H61der functions). We assume that q% : E -+ E is an adjoint vector bundle map over d/,~ for every co (i.e., q%(x): E(+~, x) ~ E(x)). We can then define the operator ~{" as before; it now acts on the Banach space C~ of e-H61der sections of E. We also define ~m = f~t(dc~ [x(dco,~)Tr c2om(x(~))%,m_~(+~mX(~O)).., c~,~x(d/o.., d?,omx(co)) where Tr is the trace on E(x(~)). Let [ r l be the norm of p~(x) for some metric on E, and I.~F I the operator on C ~ obtained by the replacement of q% by I q~ I in the definition of .~T'. Finally, let e r be the spectral radius of I~/" ]. It is easily seen from the proofs that, with these new definitions, Theorem 1.1 remains true. [For a sharper result, let [.YC"* [ be obtained by the replacement of q~'~m "" " q~'~l by I q%,, "'" ~'~1 I in ~", and take P = lim 1log ]I]JT"'~]II.] Theorem 1.3 below can similarly be extended to differentiable vector bundles. In particular, this permits the treatment of the operators jg-ct) corresponding to .;U but acting on g-forms; see Corollary 1.5. c) Let r= (r,a) with integer r/> 0 and 0~< a~ 1. We denote by C r = C'(X) the Banach space (with the usual norm) of functions X ~ C which have continuous derivatives up to order r, the r-th derivative being uniformly a-H61der. We shall write r~> 1 ifr~> 1, and Irl = r-~- ~. 1.3. Theorem. -- Let X be a smooth compact Riemann manifold. We make the same assumptions as in Theorem 1.1, but with %,, +,~ of class C', r >t 1. We require that f ~(do~) II %, II < 0% where I[" ][ is now the C" norm, and let ~g" act on C r. With these assumptions, the part of the spectrum of Og" contained in { X : ] X I > 01" I e P } consists of #olated eigenvalues of finite multiplicities. 23 178 DAVID RUELLE Define tr 3U "~ by tr.,~g"~ = f~t(d~x) ... ~(d~,,)(det(1 --D=r +,~x "'" t~,~,,)) -~ ~o.(x(~)) ~=._,(+~. x(~)) ... a,~(+~, ... +,~, x(~)) (where D, + denotes the derivative of d/ at the fixed point x), and write oo Zra d(z) = exp -- Z --trog "'~. m=l m Then, d(z) converges in (1.5) {z:lz I 01"VeP< 1 } and its zeros there are precisely the inverses of the eigenvalues of .~g', with the same multiplicities. We may therefore write (1 - z~) -~ = n(z)/d(z) where n is a holomorphic operator-valued function in (1.5). The proof of this theorem is given in Section 3. 1.4. Remarks. -- a) Theorem 1.3 also holds if we take r ----- (0, ~), = > 0, but assume that the +~, are differentiable. In that case z ~-* ~(z) d(z) is analytic and without zero in (1.4). b) The assumption that X is compact is for simplicity. It would suffice to assume that [.J,~ V,o and [.J~ qb,~ V,~ are contained in a compact subset of a finite-dimensional (non-compact) manifold. 1.5. Corollary. -- Under the conditions of Theorem 1.3, define an operator ~ acting on the space of l-forms of class C "- a on X by ~,t, f w(d~) ~e,t, l~(x) .At(T: +).~(+= x) if x e V,~, where (.~,t, dO) (x) = 0 if x r V~,. Let also tr ~ 't'" = f ~t(do~x) ... ~t(&o,,) [det(1 -- D~, +~, ... +,~.,)]-1 where Tr t is the trace of operators in At(T=c~ X) and oo Zra d~t~(z) = exp -- Z -- tro'~ It)". m~l m With these definitions ~o~ = ~, dr = d(z), and the spectral radius of ;~(~tj is <<. 0 t e r. Furthermore, if t >>. 1, the essential spectral radius of ~ct) is <~ 0 I' L+t-a e x,, dCt~( z) converges in (z: E zl 01"l+'-leP< 1} and its zeros there are precisely the inverses of the eigenvalues of jg-ct), with the same multiplidties. AN EXTENSION OF THE THEOKY OF FKEDHOLM DETERMINANTS 179 To obtain the corollary, we have to use the extension of Theorem 1.3 to vector bundles (here the cotangent bundle) as explained in Remark 1.2 b). It is clear that the spectral radius of a~U t~ is ~< 0 t e e. Note also that when t >/ 1, the degree of differentia- bility r has to be replaced by r- 1. From this, the corollary follows. (For the case where r- 1 < 1, use Remark 1.4 a).) 1.6. Corollary. -- Under the conditions of Theorem 1.3, we may write z) = II z) ] `- ''t+', where t ranges from 0 to dim X, so that the zeta funetion (1.3) is meromorphic in (1.5). This follows from the identity dlmX --0 where ~, was defined in (1.2). 1.7. Corollary. -- a) Let ~1 and .'~"2 be operators on C'1 and C's defined by the same $(do~), V,o and d/,, 9,0 of dass C "1, with rx > ra. Then, in the domain, (x: Ix I> 0 Ir'[ eP } the operators ~e" I and .~a have the same eigenvalues with the same multiplicities and the same gene- ralized eigenspaces (which consist of C'1 functions). If qb~, o?o , are C ~, it therefore makes sense to speak of the eigenvalnes and eigenfunctions of :,"g" acting on C ~, and d( z) clearly is an entire function*. b) If [~'l > OI 9 let', the elements of the generalized eigenspace of the adjoint fig" of Yg" corresponding to the eigenvalue X are distributions in the sense of Schwartz, of order s for all P - log t x I s~> I log 0 [ To prove a) note that the generalized eigenspace olaF1 maps injectively by inclusion in the generalized eigenspace of~2, but both have the same dimension given by the multiplicity of a zero of d(z). From a), one derives b) easily. 1.8. Expanding maps. -- The case where the +~ are local inverses of a map f: X -+ X has relations to statistical mechanics and applications to Axiom A dynamical systems and hyperbolic Julia sets. Various aspects of this case have been discussed by Ruelle [12], Pollicott [9], Tangerman [15], and Haydn [6], and a general review has been given in [13]. Note that the conjectures A and B of [13] are proved in the present paper. The real analytic situation, not considered here, has been discussed in Ruelle [11], Mayer [7], and Fried [3], and leads to Fredholm determinants in the sense of Grothendieck [5]. * It would be interesting to estimate the growth of d(x) at infinity. 180 DAVID RUELLE Note that an erroneous statement about the growth of determinants in [4] and [11] has been corrected by Fried [3]. For piecewise monotone one-dimensional maps see Baladi and Keller [1]. The case of an expanding map f is analysed by using a Markov partition (for which, see Sinai [14] and Bowen [2]). In the more general situation discussed here, there are no Markov partitions. Our proofs will make use, instead, of suitable coverings of X by balls. The present treatment is completely self-contained, but reference to [13] is interesting in providing for instance an interpretation of the spectral radius e P as expo- nential of a topological pressure. 1.9. Other examples. -- A class of examples where the results of the present paper apply is described as follows. Let X be a compact manifold, X its universal cover, and rc : I~ -+ X the canonical map. We assume that ~ : 32 -+ I~ is a contraction, such that d(~x, ~y) <~ O d(x,y) and that ~ : X -+ C is of class C' arid suitably tending to zero at infinity. Define (~) (x) = :~,~_,, ~(y) ~(~?~y). It is not hard to see that Js is of the form discussed above, and we have 0o Zm ~(z) = exp Z -- Z,,, ..... ,,, ~(Y,~) ... ~(Y,) ~(Ya) m~l m where the second sum is over m-tuples such that =yl = ..., =y -i = = yo, =y, = = yl. If X = R/Z and ~y -= 0y, then din(z) : d(Oz), so that ~(z) -= d(Oz)]d(z). 2. Proof of Theorem 1.1 2.1. Coverings of X by balls. -- The following construction involves the constants 0, 8 of Section 1 and a constant K which will be selected later; for the moment we only assume that 0 < K ~< 1. Let (x,)~ x be a finite (K/2) 8(1 -- 0)-dense family of points of X. In particular, the balls X, = { x :d(x, x,) < 8/2 } cover X. For each j, ~ with X~ C V~ we choose measurably u(j, ~) such that a(+~, xj, x,xj.o, ) < (K/Z) 8(1 - o) and therefore +~ Xj c X~cj, ~. For each integer m t> 0 we shall now define a finite set J~) and a family (X(a))~ j(.,) of open balls in X. We choose O' such that 8 ( O' ( 1, and we shall define j~m)and (X(a)) by induction on m, AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS 181 First, J~~ = {(i) : i ~ I }, and we let X~ = X~ be as before the balls of radius 8/2 and centers x~ = x~ forming a (*:]2)8(1- 0)-dense set in X. For m~> 1, let similarly (X~) be a finite family of open balls of radius 80"~[2 and centers x~ forming a (*:/2)8(0'--0)0'm-l-dense set in X. We put jo~, ={(i, ...,k,t) : (i, ...,k) eJ C**-1' and d(x'r,x'~-l)<~lr Choose now ,: = (I -- 0')[2. If a = (i, ..., k,g) eJ~'~ we have X~C X~ -1, and by induction X~ C ... C X~. We shall write x(a) = x~', X(a) = X~, We define p :jcm~ __~j~-l~ by p(i, ..., k, t) = (i, ..., k). Given b = (i', ...,k')ejc,,-a) and co such that XeCVo, we define v(b, co) = (i, ..., k,g) by i = u(i', co), (i, . . ., k) = v(pb, co) if m> 1, and g is chosen measurably such that a(+~ x~, -~, x~') < (,:/2) 8(0' -- 0) 0"--'. We have thus pv( (i'), co) = u(i', co), pv(b, co) = v(pb, co) for m> 1, +~ x(b) c X(v(b, co)). 9..9.. Lemma. -- We have v(b, co) E J ~"~. We write b = (i', ...,j',k'). We only have to check that d(,q, x~ ~'-~ ) <. d(xL +~ x~, .,-i) + .,-~ .< (~/2) ~(0' -- O) 0 '~*-1 + 0~0 '"-~ + (K/2) 8(0' -- O) 0"-" .< *:8(0' -- O) 0 ''-~ + *:800 ''-~ = *:80 ''-~ for m > 1, and a similar inequality for m = 1. 9..3. The operator ~. ~ We define l l if X~CV,o and i = u(j, co), ":J*(co) = 0 otherwise. Let cI),, (~cI))~ denote the restrictions of ~, DUO to X t and Xj respectively. We may then write (2.1) (a'e), (x) = X, f~(dco) x,,(co) ~(x) O,(+o x). 182 DAVID RUELLE If ~e z X~ is the disjoint sum of the X,, we may write G, C~(X,) = C'(EX,) and define an operator ~' on that space by (.K.)~ (.) = E,f~(do) .~(~0) ~(,) ,,(+, ,). This is the same formula as (2.1), but the ~, may now be chosen independently on the various X,. If we identify C~(X) with a subspace of (~, C~(X~), we see that the res- triction of ~ to C'(X) is .~s Note that (2.2) (..~"O)~, (x) -- E~ ..... ~._, f~(do,,) ... t~(do~=) .q,,,=_t(o~,,) ... ,'r ~=(x) ... %(+~ ... +~)a,~(+~ ... +~. x). 2.4. The operators .At ~. -- For m >1 1 we define an operator by the formula (2.3) (~'~',) ~ (x) = f ~(~,o~) ... ~(~) where v(j, ~) = o(o(.., v((j), o~=), ...) o~i). Define o_2": @,~i c~(x,) -~ @,~,c-~ c~(x(a)) as the restriction operator such that (02"' o)o = o, I x(a), when p=a = (i). In view of (2.2), (2.3), we have We shall also need the operator T"~' : (~.e ~1=)C'(X(a)) --+ t~)ae~(m)C'(X(a)) such that ( T'~' *)4 = *(x(a)). We define the norm on (~, G=(Xr by I *,(xl - *,(y/I~ 11 r II = max~eI (sup, [ O,(x) [ + sup**, 7/xTy / and similarly for (~aea(=)C'(X(a)). Note that, with these norms, II 02 =' [I <- 1, II T"~' I[ <-. 1. AN EXTENSION OF THE THEORY OF FKEDHOLM DF.TERMINANTS 2.5. Proposition. -- a) The spectral radius of dl (and thus .~f') is <. the spectral radius e P oflXl. b) Given r > O, we have (2.4) II .A6'*' -- .A ''~, T 'm II ~ const(0'" el'+') '' and therefore the essential spectral radius of ~ (and thus .,Y') is ~ 0 ~ e r. Using (2.2) we have 1(.~"~) (x) -- (.~"*) (Y)I a(x,y)" -< II -~'~ Iio II 9 II + const ~ I1.~ ~-' Iio II -~'-~ I1o II ~' Iio, /c-1 so that lira (11 ~" II) ~'" = lirn| (11 ~" IIo) *'~ = ~im (11 I~ I ~ 1 Ilo) ''~ ~< lima (111~ I" IIo) '~' ---- ~mo (Ilia" I" l llo) ~'' =Um (lll~ I" Ilo)', and a) follows from the spectral radius formula. Using the definition (2.3) and the estimate [l~ -- T"'~ I[0~< II~l[ (~"[2) ~, we have also I(X,-,(1 - T '=') o) (x) -- (.A"='(1 -- T ''~') O) (Y)I d(x,y)" ~< [[.ag'='[[o [[ 9 [[(~0=) ~' + const ~ C(k).[[~ [[ (~''12)" where the const comes from the H61der norm of 9 and C(k) is estimated, taking absolute values, by C(k) <. II ]./t' ]~-11 Ilo.]l I~' I'~-~ 1110 -< II IX" IH I1.111xr I=-* II. From this the estimate (2.4) follows, and b) results from Nussbaum's essential spectral radius formula [8]. 9.. 6. The operators ~', and ~". -- If k >/0, we shall define an operator .At, on ~3,~ ..... ,k, c~(x~ -" x,k) where the sum extends over the set I, of (k + 1)-tuples i = (i0, ..., ik) such that i o< ...<i~ and X~c~... c~X**4 q~. Let u(j,o~) = (U(jo, O~),...,u(j~,~)) and 1 (or--1) if Xio,...,X~kCV,o and u(j, to) is an even -rp(~) = (or odd) permutation of i, 0 otherwise. 184 DAVID RUELLE We write then Let now ~,.. ..... .k, C'(X(a,) c~ ... c~ x(~,)) be the sum over those (k + 1)-tuples of elements ofJ ~ such that X(a0) n ... n X(a,) 4= and p'~ a o = (io), ...,p'~ a~ = (ik) , with i o < ... < i k. We define then o~". O,~... ..., ,c~(x~ .... nx,,) -~O,~ o ...... ~,C~(X(ao) n .. nx(a~)) so that O~ ) is the restriction from Xlo n ... m X,~ to X(ao) n ... n X(a~). We also define ~': ~,~o ..... .~,C'(X(ao) n... n X(a~)) ~ | ..... ,,, c~(x~ n... n x,~) by (~,~ ..... ok,(+~, "'" +~. x)), where ~, and ao, 9 9 ak ~jo,) are determined as follows. If p" V(jo, ~o), . .., p" o(j~, ~) are not all different, write ~ = O. Otherwise, let ~ be the permutation which arranges these indices in increasing order, and write ~ = sign 7:, (a0, ..., as) = n(V(jo, ~), ..., v(.~, ~)). Finally, we choose an arbitrary point* x(a) ~ X(a0) n... c~ X(ak) for every (k + l)-tuple a = (ao, ..., ak) and define an operator T~ "~ on @~o ...... ,, C'(X(ao) n ... n X(ak) ) by (T~'~'~), = ~(x(a)). With these definitions we have l] O~' l[ ~< 1, 11T& =' ]l ~< 1 Note that for k = 0 the operators Jr'k, 0~"', .~r reduce to ,.~', O~ "~, ~'"~. 2.7. Proposition. -- a) The spectral radius of Jdk is <~ e P. b) Given ~> O, we have --~"~ --k [[ < const(0'" 11 ~,?, ~,,-, T,-, e ~+') and therefore the essential spectral radius of Jd k is <<, 0 ~ e P. The proof is essentially the same as that of Proposition 2.5. 2.8. Lemma. -- Suppose that ~bo~ t . .. +o,~ has a fixed point x(~) e support *?Om" Then (2.5) X,(-- 1)' X,o ..... 'm-, e 'k "% ',.-,(%.) "'" "h,',(~ "the(c%) = 1. * When k = O, take x(a0) to be the center of X(ao) as before. 185 Let I* = {j : Xj e V,~ m }and a : I* --+ I be the map such that there exist il, ..., i~_1 for which ... = 1. By assumption I* 4= ~, and clearly ~I* C I*. Let I be the set of all a-periodic points in I*, and ~ the restriction of ~ to I. Then 1 4 = e and ~ is a permutation of I. Let ~ consist of c (disjoint) cycles. Then, the non-zero terms of the left-hand side of (2.5) are those for which i o consists of the elements of/cycles of ~, with g >i 1. The value of such a term is thus (-- 1) k (-- 1)*+l-t = (-- 1) t+' and the sum is ~1>I1(-- 1) t+l (c--t)lt! c! ----1. 2.9. Corollary. -- Write (2.6) ~m,lf = El 0 ..... Ira--1 e Ik f P'(dC~ "'" ['t(dc~ (~101m_l(60CZ~) ~9o~m(X(~))) o*" (%'11]0(601) q~o~l(+r 2 9 " +ore Then = i) * 2.10. Proposition. ~ The power series 00 Z m dk(z ) : exp -- Z --~ m=lm converges for ] z [ 0 ~ e e < 1, and its zeros in this domain are the inverses of the eigenvalues of dgk, with the same multiplicities. Before proving this result, we note the following consequence. 2.11. Corollary. -- The power series 00 Z~t~ l/~(z) =exp-- X --~,, m=l m converges for [ z [ 0 ~ e r < 1, and its zeros in this domain are the inverses of the eigenvalues of J%#, with the same multiplicities. Corollary 2.9 yields 1/~(z) = H~>o [dk(z)] c-x~k. Corollary 2.11 therefore results from Proposition 2.10 if we can prove that, for [ ), [ > 0 ~ d, (2.7) ~(X) = ]~k~>0 (-- 1) * mk(X) 24 186 DAVID RUELLE where N(X) and mk(X ) are the multiplicities of X as eigenvalues of.X" and .,/d, respectively. To derive this result, let C~ = ~*o ..... ~o C~(X~ c~ ... n X~) and define coboundary operators a,:C, ~ C,+~ in the usual manner (i.e., k+l = Y, (-- 1)tr [X~t ). The existence of a C" partition of (~ r (~ ..... ~+~ t = 0 .... *~t ..... ~+~ unity associated with the covering (X,) ensures that the following is an exact sequence: r162 0 -~ c~(x) s Co ~ c, -~... ~ c~ --, c~+~ -~ ..., where [~ is the natural injection and C, = 0 for sufficiently large k. We also have ~kw, = ~,+~ ~,. Let Px=~-~ ~ resp. Pxk = ~ z --~' where the integral is over a small circle centered at X. Then, Px (resp. P~) is a linear projection of C~(X) (resp. Ck) onto the generalized eigenspace of .W (resp. ~'k) corresponding to X. Furthermore P~+~ % = %Px,. P~o ~ = ~Px, We therefore have an exact sequence 0 -+ im Px ->~ im Pxo -~ im Pxl --> ... ---> 0 so that dim im Px =- ~k>~0 (-- 1) ~ dim im Px, which is precisely (2.7). 2.12. Proof of Theorem 1.1. -- Theorem 1.1 follows from Proposition 2.7 and Corollary 2.11. We are thus left with Proposition 2.10 to prove. 2.13. Proof of Proposition 2.10. -- There is a finite number of eigenvalues X~ of~'~ such that [ Xa [ > 0 '~ e e. If mj is the multiplicity of Xj, we may write Y~a mj(Xj) '~ ----- ~ )'7 ]~v a~v(S~v) = ~Jv a~v(..g/~ S~v) where (%v) and (Sjv) are dual bases of the generalized eigenspaces of ..~'~ and .Me, res- pectively for the eigenvalue Xa. Therefore where Car has the constant value S~r(x(a)) on X(a0) c~ ... n X(a~). Using Proposition 2.7 we have (2.9) I X,v ,,((~'~'~' -- .~'L '~' T~ '~') O~' S,~) [ ,< const(0'" el'+') '~. Let X, be the characteristic function of X(ao) n ... n X(al, ) as an element of | ..... o,, c'(x(~o) n ... n X(a~)). AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS 187 Then (2.10) 23~v %v(.Jt'~ =' Cjv)" = Zjv Y~, Sjv(x(a)) %vk~* k/~"' Z,) = X,((1 - ~) zc,m, v_~ z.)(x(,,)) where #~ is the projection corresponding to the part of the spectrum of ..r k in {x: I xl.< 0'-~'). The right-hand side of (2.10) is the sum of two terms. The first can be written as Z,(X';"' Z.) ('~(")) = Xc~ ..... 'k' ~ Xk Z.o: .".o = ~, "'" Z.k: =".k = ~k f ~t(&%) ... [x(do~.,) 2~,, (sign ) 8((ao, ..., ak) , n(v((io) , ~), ..., v((ik) , G))) ~om(X(a)) ... q~,~1(+~,2 ... +comX(a)) = ZiO ..... ,m_l~ikf~L(d(.o1) ... [s *loim_l(fDm) . . . "Cl211(fD2) "t'ilio(0.)l) q~'om(X(] v(i0' ~)])) "'" ?'~x(+'~, "'" +~=X([ v(io, ~)1)) where Iv(i, ~)[ is the permutation of (V(io, ~o), ..., v(ik, ~)) such that p'~ [ v(i, co)] -- i. If we replace in the right-hand side x(I v(i0, ~)]) by the fixed point x(~) of +,~a ... +,~=, the error is bounded by const(0 '" eP+') m (using the same sort of estimates as in the proof of Proposition 2.5). Therefore, by the definition (2.6) of ~,,~, we have (2.11) ] ]~.(dt'~ '~) X,) (x(a)) -- ~,~ [ ~< const(0 '~ eP+") '~. We are left with the study of X.(~', ~' z.) (x(~)). Remember that the sum is over the set J~'~ of those a = (a0, ..., ak) ~ (J"~)*-~ such thatp"a= (io, ...,i,) with i 0< ... < i k. Note that, if0~<g~< m, we may write X&~J~rn)(~r m) )(at) (X(P m-I a)) = Xb~kt)(~J~c2--l*/~c~t)Xb) (X(b)) Clump together those a such that p,~-t a = b). Therefore ~&(~tn) ~) (X(a.)) -- ~-]~! EIk( #~*~r ~..1) ix(i)) = ~ y~.((~,x':', z.) (x(f"-' ..)) - (~,x';", z.) (x(p "-'+~ ,.))) /=1 = ~ Y~,,((~7-t~ff' z~)(~(b)) -(~.-'.~', z~)(x(~b))). /=1 From this we get, using (2.13) below, (2.12) I ]E,(~-A'~ ") 7~) (x(a)) I ~< const II ~" I1 + const 2~ II ~-t ll.~b~a~,, [[..g[~,, Xb II.d(x(b), x(pb)) ~ /=1 ~< const [(O"e~+~) " + Z (O"ee+*)'~-t(eX'+')to't~] t=l <. const m(O '~ ee+~) ". 188 DAVID RUI'LLE Putting together (2.8), (2.9), (2.10), (2.11), (2.12) we obtain I ]~J mj(X~)" -- ~.~ I < const m(0 '~ eP+') '~ and therefore 00 ~m log( d~( z) /l'I j(1 -- X s z)'~i = Z -- ( Ej mj(~.~) '~ -- ~.~) m~l m converges for I z I 0'~ e~+* < 1, proving Proposition 2.10. In deriving (2.12) we have used the inequality (2.13) ]g, ea~ t'II ~L" z, II -< const(eP+") t which we shall now prove. Given [3 > 0 we set ~,~ = [ % ] -t ~ 11% I}" In the definition--Section 2.6---of dt'~ t~ if we replace q0,~ by q~,~ and suppress the factor r = 4- 1 we obtain an opetaror ~,~t) (Mk~ r (x) .... tt, f tz(d%) 9 bt(do~t) ~t~(*) -" ~.~(~,"" +~t~)%0 ..... .,,(%.'" +~t'-) where (a0,..., a~) is a permutation of (V(jo, d),..., v(jk, G)). In particular II-~"~" z, I1o -< I1 "~'" If x,y E Xjo n... r~ Xjk , we also have Zb), (x) -- (...~" Z,), (Y)[ < jtz(d%) ... ~(doat) %or(x) "'" %~,(+,~, "" +,or x) -- *O,~t(Y) "'" %,,(+0,, "" +otY) l <~ Y~ ~*(d%) ... tz(d%) ep,~t~(x) ... ~,~i+,~(+oi+, "" ~bot x) 'f ~,~('+,~,§ " " +o,t x) -- ~o,,(+o,+, . . . +o, t Y) [ %,-t~(~b~,i"" +,~tY) "'" %,~(+,o,... +,~tY)" where the integrals are restricted to those (cot, ..., ~ for which b is a permutation of (V(jo, ~), ..., v(L, ~)). We may write I %(+~,, ... +~tx) - ~(+~, ... +~tY)l ~< I[ ~~ II ( Or-' d(x,Y)) ~ <~ const ?oip(hb,~i., ... ~b,~ t x) 0 "r176 d(x,y)" and similarly %~(+~,, "" +-tY) "< %d+ .... ,-.. +,or x) (1 + const o'~t-"). (2.14) Therefore I(,-g~ t' zb)j (x) - (,~t, z~), (Y)I ~< const t ]1 x,fm 9 ,~ z, Iio d(x,y)" hence (2.15) II .~,L~, z, II < c(p).e II ~,~,,, 9 .-~ z, I[o. AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS 189 From (2.14) we also obtain ~'*~ zb)j (x) where C'(~) does not depend on/. Therefore 9 .~ z~ IIo< c'(~) E| sup= I(Mkt~ 1)j (x)l .< C"(f~) II M~% Ill0 and with (2.15) this gives Y~.~r II ~,t, z~ II -< o(p) c"(p) e II -.-k~r 1 ]lo ~ C'"(~) (eX~'+"~) t where e ~'cp~ is the spectral radius with respect to the [] []0 norm of the operator .At'p obtained if we replace %, by q~,~p in the definition of dr'. Note that .At'~ is close to [ .At' [ for ~ small: I1~'~- I~ I II0- < ~ f~(d~)II ~ II. Using the upper semicontinuity of the spectral radius we may thus choose [~ such that i.e., (2.13) holds as announced. 3. Proof of Theorem 1.3 3.1. The essential spectral radius of o~g'. -- We shall follow the proof of Theorem 1.1 in Section 2, and note what changes have to be performed to deal with the differentiable situation. First of all, we make a choice of charts for the balls X,, which will thus be identified in what follows with subsets of Euclidean space. We may assume that the bails X, have small radii and that the Riemann metric is closely approximated by the Euclidean metric. Confusion between the two metrics is then inconsequential. The linear structures which we have chosen will allow us to define Taylor expansions. Replacing C ~ by C v everywhere, we define r162 ~"", O~"', Mt'~, ..#t'~"', O~' as before. The operator T ~'~ on @,cac,,~ C'(X(a)) is now defined by (T r *), = Taylor expansion of order r of * at x(a) and similarly for --kTr We have then II r - T':',X, Iio.< const II r II 0.-,,l. Following the arguments of Sections 2.5, 2.6, 2.7 with obvious changes, we get (3.1) II-r - ..gt'~', T~" ]l ~< e~ 0'l'l er+') " and therefore the essential spectral radius of dt'~ is ~< 0 Irl e P. In particular, the same estimate holds for the essential spectral radii of .At' and 2U. 190 DAVID RUt"LLE 3.2. Proposition. -- Define Tr.~r =/~(dox) ... vt(do,~) (det(1 -- D.,~, +,o~ ... +,,,))-x (~._~(~o.) ~.(x(~))) ... (%~(,o~) ~(+~, ... +~, x(~))). Then, the power series oo Z m d~~ = exp -- Y. -- Tr.Ar m=lm converges for [ z ] 0 l" I e v < 1, and its zeros in thb domain are the inverses of the eigenvalues of .~k, with the same multiplicities. Before proving this result, which corresponds to Proposition 2.10, we note that it allows us complete the demonstration of Theorem 1.3. We have indeed a(z) = II,~> o (a?'(z)) ~-~ by Lemma 2.8. The proof of Corollary 2.11 again applies, and yields that the zeros of d(z) in (1.5) are precisely the inverses of the eigenvalues of.ft', with the same multiplicities. 3.3. Remark. -- Before embarking in the demonstration of Proposition 3.2, we 0" prove a necessary estimate. Let n = (nl, ..., na~x) be a multi-index, ~ the corres- ponding derivative, and n! = nil ... hoax!. We assume that ] n [ = nl + ... + na~x-< r. Define then E (") ~x~S = E/o , .-., im-x f~L(Ks176 ...... ~L(d~m) 'I'~O,m_t(O)m) '~(110(s and assume that ~ e X(v(i 0, g)) for k 1, 2, 3. Replace in v~,~ the expression 0- Ox;(...) by its Taylor expansion around ~3(i0, ~), keeping derivatives of total order up to r, and then put x = ~(io, ~). The error thus made is bounded by const (0,,~)lr 1--I-I. (ev+*)= (0',~)l-[ = const(eP+. 0' Irl),~. Define now E~ ----- 2~.:[.[~<, ~.. E~ ~ i.e., (3.2) = z.:l.,.<. ..... ... [~(&o,~) v~o,,,_x(%, ) ... *,x~(Ol) 1 0" ,! 0x" (~gx) "" ~I(+~,'-" +~x)(%... +~x- ~(i0, ~))")1._~,~.~,. AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS 191 Introducing limited Taylor expansions as explained above in each term of E~,, we simply obtain Et,. Therefore (3.3) [E~x -- E~, 1~< const(d +* 0'lrl)% 3.4. Proof of Proposition 3.2. -- We shall prove the proposition for ,~'(= M(0) rather than ~r (This simplifies notation, and the general case is easily recovered by reference to Section 2.13.) There is now a finite number of eigenvalues Xj of ~r such that ] Xj ] > 0 'ltl e P. Let mj be the multiplicity of X~, and (%v), (S~v) be dual bases of the generalized eigen- spaces of.~r and ~r respectively for the eigenvalue Xi" Then (3.4) Zt m~(X~)" = Zjv %v(~ r S jr ) = Zjv (~t,((.~ '(") -- .At '(=' T '=') Q~") Sn) + Zjv (~n(..s ('~) Cjv ) where C~v [ X(a) is the Taylor expansion to order r of Sjv at x(a). Note that (3.1) gives (3.5) ] ~v %v(('~'('~) -- "A~'(") T(')) O~') S~v) ~< c~ el'+=)=. Let ~ denote the characteristic function of X(a) and write 0~ for the derivative of order n = (nl, ..., n~,..x) evaluated at 4. We have 9 " s,~ ~,.(.~' ((. - ~(~))" ~)) (3 6) Z~, ~.(~("~' C~,) = Z~.~ Z~ Z.:I.I.<,~ " a=(., "' = Z= Z,:I,I~< , n5 0"~(='((1 -- ~) "~"')((" -- x(a))'* Z,)) the part of the spectrum of ~ in where ~ is the projection corresponding to {X: ]k[~< 0'lrl eZ'}. Further, 1 2. ~,,,.,. _ x(a))" ~) (3.7) Z~ Z. ~.. v.,~ ,,. = Z= Z. ~. ~(a,ol) ... ~(a,~) ~(a, v(p" a, ~)) ax-- ~ (~.(x) .. ~=~(+=, ... +~. x) (+=~ ... +=. x - ~(=))')I.-.,., l 0 Z. ~. F~" (~==(x) ... ~=~(+~, ... +~, x) (+0,... +0= 9 - x(~(io, ~)))")I.-=,.,~,~,, = E~I where we have used the notation (3.2) with ~t(io, ~) = x(v(io, ~)). If we choose ~(i0, ~) = x(~), we get E~ s l- < const(O 'lrl d+') '~ (3.8) I Er - 192 DAVID RUELLE in view of (3.3). Furthermore, since x(~) is a fixed point of +,~ ... d&~=, ~..(x(~)) .-. ~o,(+., ... +~. x(~)) ~..i.,-<, ~ ~.. where iF, is ~ polynomial of order [ n [ in the elements of the matrix D,~)(~b,~ 1 ... ~bo,,) of derivatives at x(~), and ~tt',jnl is invariant under linear changes of coordinates. It is easily recognized that ~tF,Jn! is the development of (det(1 -- D| ~1 ... ~,~m)) -1 to order r (take D to be in Jordan normal form). Therefore (3.9) I E~, -- tr.~'m [ ~< const(0 '+a er+') ". From (3.7), (3.8), (3.9) we get 1 I ..g(,.~((. ~< const(O '1'1 er+') =. (3.10) Zo Y~. ~. o~., " -- x(a))" X~) -- tr'~r I There remains to estimate Note that, if 0 <. t <~ m, we may write :Z.~ jc., :Z. ~. o2,,.-t., ~""((- - x(p "-t a))" z,) Thus z. r x, ,,/"z,/ -- ~ 52~za(t~ Y~n 1 ~dlm_tjlr ( - ,-1 ~. (o,%) 9 - x(b))" x~) _ ~. ~.~,.-t~,,~,((. _ x(pb))" z~)). The absolute value of the right-hand side can be estimatcd in terms of Taylor expansions (as in Remark 3.3). Using also (3.12) below, we get a bound const ~ ]~be#t, Z, ~ d(x(b), x(pb)) I'l-I"l ,-a II ~r--' II. II at,m((. - x(O))" z~)II co~st :~ Z. 1 (o,,)1,.-1-1. (o,l,I :+.),--,. (a'+')' (o,')l-. t-1 n! = const m(O '1"1 el'+=) ~. AN EXTENSION OF THE THEORY OF FREDHOLM DETERMINANTS 193 Therefore Y., ~, ~. 0~"c, , ~Mr -- x(a))" 7.,) <- const m(0 '1"1 eP+') ". (3.11) From (3.4), (3.5), (3.6), (3.10), (3.11) we conclude that [ 2~ mj(;~j)" -- tr~r [ ~< const m(0 'lrl er+') ". Therefore co Zr~ log(d~ ~ (z)/Hj(1 - X s z)"/) = Y~ -- (]~rnj(X~)" - tr..//") ra=l 7n converges for I z I 0' I r I e a" +, < 1, proving Proposition 3.2. We have used the inequality (3.12) Y~bc a 't) II Ml't)((" -- x(b))" ;(b l] ~< c~ t (0't) I"1 which is proved like (2. I3). REFERENCES [1] V. BALADt and G. KELLER, Zeta functions and transfer operators for piecewise monotone transformatiorm, Commun. Math. Phys., 12'1 (1990), 459-477. [2] R. BOWEN, Markov positions for Axiom A diffeomorphisms, Trans. Amer. Math. 8or 154 (1971), 377-397. [3] D. FRmD, The zeta functions of Ruelle and Selberg, I., Ann. Sci. E.aV.S., 19 (1986), 491-517. [4] A. GROTVlENmECK, Produits tensoriels topologiques et espaces nuclfaires, Memoirs of the Amer. Math. Sot., 16, Providence, RI, 1955. [5] A. GROT~ENI>I'~CK, La th6orie de Fredholm, Bull. Soc. Math. Frame, 84 (1956), 319-384. [6] N. HAYDN, Meromorphic extension of the zeta function for Axiom A flows, Ergo& Th. and Dyaam. Syst., 10 (1990), 347-360. [7] D. MA'~SR, On a ~ function related to the continued fraction transformation, Bull. Soc. Math. France, 104. (1976), 195-203. [8] R. D. NUSSBAUM, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478. [9] M. POLLICOTT, A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergo& Th. and Dynam. Syst., 4 (1984), 13.5-146. [10] F. R.iEsz et B. Sz-NAOY, Lefons d'analysefonctionnelle, 3* ~d., Acaddmie des Sciences de Hongrie, 1955. [ 11 ] D. RtmLLE, Zeta-functions for expanding maps and Anosov flows, Inventiones Math., 34 (1976), 231-242. [12] D. RU'~LLE, Thermodynamic Formalism, Encyclopedia of Math and its Appl., 5, Addison-Wesley, Reading, Massa- chusetts, 1978. [13] D. RIW.I.t.E, The thermodynamic formalism for expanding maps, Commun. Math. Phys., 125 (1989), 239-262. [14] Ia. G. SINAI, Construction of Markov partitions, Funkts. Analiz i ego Pril., 2, No. 3 (1968), 70-80. English trans- lation: Functional Anal. Appl., 9. (1968), 245-253. [15] F. TANOERMAN, Meromorphic r of Ruelle zeta functions, Boston University Thesis, 1986 (unpublished). Institut des Iiautes Etudes Scientifiques 35, route de Chartres 91440 Bures-sur-Yvette Manuscrit re~u le 16 janvier 1900.
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Published: Aug 31, 2007
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