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Return times of dynamical systems

Return times of dynamical systems APPENDIX by ]. Bourgain, H. Furstenberg, Y. Katznelson, D. S. Ornstein Let (X, iS, V, T) be an ergodic system and let A E ~ be of positive measure Ez(A) > 0. For x ~ X, consider the return time sequence A, = { n e Z+ [ T" x e A }. By Birkhoff's pointwise ergodic theorem, the sequence A~ has positive density for v-almost all x e X. This fact refines the classical Poincar6 recurrence principle (cf. [Fu]). An even stronger statement is given by the Wiener-Wintner theorem: there is a set X' of X of full measure such that the sums -- ]~ zx(T"x) z" converge for all z in the unit circle Ct = { z e C ] [ z I ---- l } and x e X'. Thus from general theory of unitary operators, this fact may be reinterpreted by saying that almost all sequences A~ satisfy the L ~, hence the mean ergodic theorem. Our purpose here is to prove the following fact, answering a question open for some time. Theorem. -- With the notation above, A~ satisfies almost surely the pointwise ergodic theorem, i.e., the averages -- ~ S" g N x~,<~ nGAx converge almost surely for any measure preserving system (Y, ~, v, S) and g e L:(Y). The argument given next actually yields a more precise condition on the point x. Letf ~ L~176 be obtained by projecting ZA on the orthogonal complement of the eigenfunctions of T. It clearly suffices to prove that for almost all x e X, {f(T" x) } is a " summing sequence ", i.e., (*) ~ a.<Y~..<~ f(T" x) g(S"y) -+ 0 a.e. y ~ Y for any measure preserving system (Y, ~, v, S) and g e L| (The contribution of the eigenfunctions is taken care of by Birkhoff's theorem.) Observe the equivalence of the following statements: (i) f has continuous spectral measure, (ii) < T"f,f > = ~1(nl, ~ a continuous measure, lq (iii) (l/N) Y~f(T"x)f(T"~) .-+0 a.e. in (x,~) as N -->oo. 1 POINTWISE ERGODIC THEOREMS FOIl. ARITttM-ETIC SETS 4S Proof of (ii) :> (iii).- Write F = lim (l/N) Y.f(T" x)f(T" ~), a limit which exists by the ergodic theorem, and 1] F ][' = lim (I/N") ~ (~(n -- m)) 2 ---- O. Proposition. --Assume x generic for f and (I/N)2gf(T"x)f(T" 4) -+0, a.e. in ~ (/). Then {f(T" x)} # a summing sequence. Proof. ~ I) Assume that for some (Y, N, v, S) and g E L ~ (Y) there is a set B" of positive measure for which the limsup of (*) is positive. Then there exists a > 0, B C B ~ ~(B) > 0 and a sequence of intervals R~ = (Lj, M~) (called " ranges ") such that for everyy e B and everyj there exists nj e IR~ (n~ = nj(y)) such that ni (**) >1 f(T" x) g(S"y) > anj. II) Given 8 > 0, there exists K = K(N, 8) such that ~(U S'B) > 1 --8. lr III) Write ? for the indicator function of [J S ~ B. If Mo is large enough, and if we denote by G the set G ---- {y : [ (1In) ~ ~(SJy) -- 1 [ < 2 8 for all n > M0 }, then v(G) > 1 -- 8. x IV) For notational convenience we assume thatf has finite range, and we denote by B, the set of all n-blocks for f, i.e., the set of words w~ ") --= (f(T ~+1 x), ...,f(T k+" x)); w~ ") appears with density p(w~")). Given 8 > 0 (~ can be chosen once and for all as a function of a and v(B) in I)) let Na be such that for each set A~C X, r > 1 -- 8, I(1/N) Y,f(T"x)f(T" 4)] < 8 for all ~ EA n and N> Nn (of. assumption (!)). Given a range (L, M) with E > N n, set N ---- N(M) so that in any interval on the integers of length >I N the statistics of the n-blocks (for f) with n ~< M is correct. Denote by B~, the n-blocks that have the form (f(T~), ...,f(T" 4)) with ~ E A a (we are interested in n e (L, M)). For L < n < M the total probability (---- density) of the blocks in B~ exceeds 1 -- 8 (in any interval of length >/N(M)). Notice also that heads of M-blocks which are in B~ are in the appropriate B~,. V) A sequence of ranges {(L~, M~)} is properly spaced if L~+ a > N(M~). (We also assume L~ > N~. Another assumption on L~ is that it is > M0 (recall the definition of G in III) and assume that K (II)) is ,~ L~.) Going back to I), we select a properly spaced sequence of ranges { (L~, M#)}~= ~ (J depending on a) and N large enough so that N ~ N(Mj). Recall B from I) and G from III). 44 JEAN BOURGAIN For anyy e B c~ G we define a sequence { c,(y)}~_ 1 which is a sum of J sequences (layers) { c~(y)} having the following properties: (0t) For all j, n andy, c~(y) is in the range off (in particular uniformly bounded) ([~) Vorjx 4=j,, I(1/N) Y~ c~l(y) d,*(y)[ < 8 (y) (l/N) 2~ cJ.(y) g(Sny)>a--8, j= 1,...,J n=l (0~) and (~) together imply [(l/N) Y,(c.(y))2] x/2 ---- O (~v/j + 8J), and (y) implies (l/N) Y, c.(y)g(S"y) > J(a - 8). Contradiction. We construct { c~. } in reverse order on j. The number d~(y) is defined as follows: tx(y ) is the first index k > 0 such that S*y ~ B; on the interval (tl(y), tx(y) + na(Stx~Y~y)) we set c,a(y) = f(T" - t~,,, x), t2(y ) is the index of the first point in the S-orbit ofy after tx(y ) + na(Stx~'~y) which is in B, and on the interval (t2(y), t~(y) + ha(St'Or)y) we copy again {f( Tk -J~ a etc. The intervals on which we copy those starting n a blocks fill most of [1, N]. We refer to these as the basic intervals of the J-layer. Outside of these, set ca.(y) arbitrarily. We now define ca,-~(y) in a similar manner within every basic interval of the J-layer, with the additional restriction on the starting place of the new basic blocks that (in addition to the fact that the corresponding point in the orbit ofy is in B) the matching piece of the basic J-layer block in is B*, i.e., more or less orthogonal to the " new " basic block; see IV). Since the " orthogonal " blocks have density > 1 -- 8, the new basic blocks cover more than 1 -- 3 8 of [1, N]. We continue with ca-~(y), . .., c~.(y), working each time within the basic blocks of the previous level and introducing blocks which are " orthogonal " to all previous levels. Remarks. lq (i) The condition that (l/N) ]~f(T" x)f(T" ~) ~0 a.e. in ~(!) is a special case of (*) and hence necessary. One can construct examples showing that it is not a consequence of the genericity of x. (ii) One may construct a sequence A = { k. }, k, = o(n), and a weakly mixing system (Y, S) such that (l/N) Y~g(S*"y) does not converge a.e., for some g ~ L~~ (This question was considered in [Fu], p. 96.) REFERENCES J. ]30UROAIN, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math., 61 (1) (1988), 39-72. J. BOtmOAIN, On the pointwise ergodic theorem on LP for arithmetic sets, ibid., 73-84. JkX"~st'(')~= POINTWISE ERGODIC THEOREMS FOR ARITHMETIC SETS 45 J. BOURGAIN, An approach to pointwise ergodic theorems, Springer LNM, 1317 (1988), 204-223. [B,] J. Return time sequences of dynamical systems, preprint IHES, 3/1988. [Bs] J. Bouao~aN, Almost sure convergence and bounded entropy, Israel J. Math., 68 (1) (1988), 79-97. [B,] J. BouRG~u% Temps de retour pour les syst~mes dynamiques, GRASc Paris, Ser I, 806 (1988), 483-485. [Fu] H. Recurrence in ergodir theory and combinatorial number theory, Princeton UP, 1981. [Ga] A. GA~XA, Martingale Inequalities, Benjamin, 1970. [K-W] Y. I~TZ~mLSON, B. Wsms, A simple proof of some ergodic theorems, Israel J. Math., 42 (4) (1982), 391-395. [L6] D. LgPrNoT~, La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 86 (1976), 295-316. [Ma] J. M. MAR~a?._n~D, On Khinchine's conjecture about strong uniform distribution, Proc. London Math. Sot., 21 (1970), 540-556. [Nev] J. NEwu, Martingales d temps distret, Masson, 1972. [Ra] D. A. RAIKOV, On some arithmetical properties of summable functions, Mat. Sb., 1 (43) (1936), 377-384. [Ri] F. RiEsz, Sur la th6orie ergodique, Comment. Math. Helv., 17 (1945), 221-239. R. SAL'~M, Collected works, Hermann, Paris, 1967. IS] E. STSXN, On limits of sequences of operators, Ann. Math., 74 (1961), 140-170. [so [Vaug] R. C. V^UOHAN, The Hardy-Littlewood method, Cambridge Tracts, 70 (1981). I. M. VmOORADOV, The method of trigonometrical sums in the theory of numbers, Interscience N. Y., 1954. Win] [w] B. W,,iss, Private communications. [W1] M. Wmm)L, Pointwise ergodic theorem along the prime numbers, Israel J. Math., 64 (1988), 315-336. j. B.: Institut des Hautes ]~tudes Scientifiques 35, route de Chartres 91440 Bures-sur-Yvette France H. F.: Hebrew University 91904 Jerusalem Israel Y. K. and D. O.: Department of Mathematics Stanford University Stanford, California 94305 ]~tats-Unis Manuscrit refu le BO dgcembre 1988. FtraSTENB,~RO, BOUROAXN, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © 1989 by Publications Mathématiques de L’I.É.E.S.
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN
0073-8301
eISSN
1618-1913
DOI
10.1007/BF02698839
Publisher site
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Abstract

APPENDIX by ]. Bourgain, H. Furstenberg, Y. Katznelson, D. S. Ornstein Let (X, iS, V, T) be an ergodic system and let A E ~ be of positive measure Ez(A) > 0. For x ~ X, consider the return time sequence A, = { n e Z+ [ T" x e A }. By Birkhoff's pointwise ergodic theorem, the sequence A~ has positive density for v-almost all x e X. This fact refines the classical Poincar6 recurrence principle (cf. [Fu]). An even stronger statement is given by the Wiener-Wintner theorem: there is a set X' of X of full measure such that the sums -- ]~ zx(T"x) z" converge for all z in the unit circle Ct = { z e C ] [ z I ---- l } and x e X'. Thus from general theory of unitary operators, this fact may be reinterpreted by saying that almost all sequences A~ satisfy the L ~, hence the mean ergodic theorem. Our purpose here is to prove the following fact, answering a question open for some time. Theorem. -- With the notation above, A~ satisfies almost surely the pointwise ergodic theorem, i.e., the averages -- ~ S" g N x~,<~ nGAx converge almost surely for any measure preserving system (Y, ~, v, S) and g e L:(Y). The argument given next actually yields a more precise condition on the point x. Letf ~ L~176 be obtained by projecting ZA on the orthogonal complement of the eigenfunctions of T. It clearly suffices to prove that for almost all x e X, {f(T" x) } is a " summing sequence ", i.e., (*) ~ a.<Y~..<~ f(T" x) g(S"y) -+ 0 a.e. y ~ Y for any measure preserving system (Y, ~, v, S) and g e L| (The contribution of the eigenfunctions is taken care of by Birkhoff's theorem.) Observe the equivalence of the following statements: (i) f has continuous spectral measure, (ii) < T"f,f > = ~1(nl, ~ a continuous measure, lq (iii) (l/N) Y~f(T"x)f(T"~) .-+0 a.e. in (x,~) as N -->oo. 1 POINTWISE ERGODIC THEOREMS FOIl. ARITttM-ETIC SETS 4S Proof of (ii) :> (iii).- Write F = lim (l/N) Y.f(T" x)f(T" ~), a limit which exists by the ergodic theorem, and 1] F ][' = lim (I/N") ~ (~(n -- m)) 2 ---- O. Proposition. --Assume x generic for f and (I/N)2gf(T"x)f(T" 4) -+0, a.e. in ~ (/). Then {f(T" x)} # a summing sequence. Proof. ~ I) Assume that for some (Y, N, v, S) and g E L ~ (Y) there is a set B" of positive measure for which the limsup of (*) is positive. Then there exists a > 0, B C B ~ ~(B) > 0 and a sequence of intervals R~ = (Lj, M~) (called " ranges ") such that for everyy e B and everyj there exists nj e IR~ (n~ = nj(y)) such that ni (**) >1 f(T" x) g(S"y) > anj. II) Given 8 > 0, there exists K = K(N, 8) such that ~(U S'B) > 1 --8. lr III) Write ? for the indicator function of [J S ~ B. If Mo is large enough, and if we denote by G the set G ---- {y : [ (1In) ~ ~(SJy) -- 1 [ < 2 8 for all n > M0 }, then v(G) > 1 -- 8. x IV) For notational convenience we assume thatf has finite range, and we denote by B, the set of all n-blocks for f, i.e., the set of words w~ ") --= (f(T ~+1 x), ...,f(T k+" x)); w~ ") appears with density p(w~")). Given 8 > 0 (~ can be chosen once and for all as a function of a and v(B) in I)) let Na be such that for each set A~C X, r > 1 -- 8, I(1/N) Y,f(T"x)f(T" 4)] < 8 for all ~ EA n and N> Nn (of. assumption (!)). Given a range (L, M) with E > N n, set N ---- N(M) so that in any interval on the integers of length >I N the statistics of the n-blocks (for f) with n ~< M is correct. Denote by B~, the n-blocks that have the form (f(T~), ...,f(T" 4)) with ~ E A a (we are interested in n e (L, M)). For L < n < M the total probability (---- density) of the blocks in B~ exceeds 1 -- 8 (in any interval of length >/N(M)). Notice also that heads of M-blocks which are in B~ are in the appropriate B~,. V) A sequence of ranges {(L~, M~)} is properly spaced if L~+ a > N(M~). (We also assume L~ > N~. Another assumption on L~ is that it is > M0 (recall the definition of G in III) and assume that K (II)) is ,~ L~.) Going back to I), we select a properly spaced sequence of ranges { (L~, M#)}~= ~ (J depending on a) and N large enough so that N ~ N(Mj). Recall B from I) and G from III). 44 JEAN BOURGAIN For anyy e B c~ G we define a sequence { c,(y)}~_ 1 which is a sum of J sequences (layers) { c~(y)} having the following properties: (0t) For all j, n andy, c~(y) is in the range off (in particular uniformly bounded) ([~) Vorjx 4=j,, I(1/N) Y~ c~l(y) d,*(y)[ < 8 (y) (l/N) 2~ cJ.(y) g(Sny)>a--8, j= 1,...,J n=l (0~) and (~) together imply [(l/N) Y,(c.(y))2] x/2 ---- O (~v/j + 8J), and (y) implies (l/N) Y, c.(y)g(S"y) > J(a - 8). Contradiction. We construct { c~. } in reverse order on j. The number d~(y) is defined as follows: tx(y ) is the first index k > 0 such that S*y ~ B; on the interval (tl(y), tx(y) + na(Stx~Y~y)) we set c,a(y) = f(T" - t~,,, x), t2(y ) is the index of the first point in the S-orbit ofy after tx(y ) + na(Stx~'~y) which is in B, and on the interval (t2(y), t~(y) + ha(St'Or)y) we copy again {f( Tk -J~ a etc. The intervals on which we copy those starting n a blocks fill most of [1, N]. We refer to these as the basic intervals of the J-layer. Outside of these, set ca.(y) arbitrarily. We now define ca,-~(y) in a similar manner within every basic interval of the J-layer, with the additional restriction on the starting place of the new basic blocks that (in addition to the fact that the corresponding point in the orbit ofy is in B) the matching piece of the basic J-layer block in is B*, i.e., more or less orthogonal to the " new " basic block; see IV). Since the " orthogonal " blocks have density > 1 -- 8, the new basic blocks cover more than 1 -- 3 8 of [1, N]. We continue with ca-~(y), . .., c~.(y), working each time within the basic blocks of the previous level and introducing blocks which are " orthogonal " to all previous levels. Remarks. lq (i) The condition that (l/N) ]~f(T" x)f(T" ~) ~0 a.e. in ~(!) is a special case of (*) and hence necessary. One can construct examples showing that it is not a consequence of the genericity of x. (ii) One may construct a sequence A = { k. }, k, = o(n), and a weakly mixing system (Y, S) such that (l/N) Y~g(S*"y) does not converge a.e., for some g ~ L~~ (This question was considered in [Fu], p. 96.) REFERENCES J. ]30UROAIN, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math., 61 (1) (1988), 39-72. J. BOtmOAIN, On the pointwise ergodic theorem on LP for arithmetic sets, ibid., 73-84. JkX"~st'(')~= POINTWISE ERGODIC THEOREMS FOR ARITHMETIC SETS 45 J. BOURGAIN, An approach to pointwise ergodic theorems, Springer LNM, 1317 (1988), 204-223. [B,] J. Return time sequences of dynamical systems, preprint IHES, 3/1988. [Bs] J. Bouao~aN, Almost sure convergence and bounded entropy, Israel J. Math., 68 (1) (1988), 79-97. [B,] J. BouRG~u% Temps de retour pour les syst~mes dynamiques, GRASc Paris, Ser I, 806 (1988), 483-485. [Fu] H. Recurrence in ergodir theory and combinatorial number theory, Princeton UP, 1981. [Ga] A. GA~XA, Martingale Inequalities, Benjamin, 1970. [K-W] Y. I~TZ~mLSON, B. Wsms, A simple proof of some ergodic theorems, Israel J. Math., 42 (4) (1982), 391-395. [L6] D. LgPrNoT~, La variation d'ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 86 (1976), 295-316. [Ma] J. M. MAR~a?._n~D, On Khinchine's conjecture about strong uniform distribution, Proc. London Math. Sot., 21 (1970), 540-556. [Nev] J. NEwu, Martingales d temps distret, Masson, 1972. [Ra] D. A. RAIKOV, On some arithmetical properties of summable functions, Mat. Sb., 1 (43) (1936), 377-384. [Ri] F. RiEsz, Sur la th6orie ergodique, Comment. Math. Helv., 17 (1945), 221-239. R. SAL'~M, Collected works, Hermann, Paris, 1967. IS] E. STSXN, On limits of sequences of operators, Ann. Math., 74 (1961), 140-170. [so [Vaug] R. C. V^UOHAN, The Hardy-Littlewood method, Cambridge Tracts, 70 (1981). I. M. VmOORADOV, The method of trigonometrical sums in the theory of numbers, Interscience N. Y., 1954. Win] [w] B. W,,iss, Private communications. [W1] M. Wmm)L, Pointwise ergodic theorem along the prime numbers, Israel J. Math., 64 (1988), 315-336. j. B.: Institut des Hautes ]~tudes Scientifiques 35, route de Chartres 91440 Bures-sur-Yvette France H. F.: Hebrew University 91904 Jerusalem Israel Y. K. and D. O.: Department of Mathematics Stanford University Stanford, California 94305 ]~tats-Unis Manuscrit refu le BO dgcembre 1988. FtraSTENB,~RO, BOUROAXN,

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Published: Aug 30, 2007

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