autism 3—:- ‘ “VT-h}; 1'. . . n-lm. —. . n A u #7::- ”up ~ u.,‘.;‘ "-31. n.4,. #1] "a f; w. {5' w.- 4:! VI' mas (3&03 S‘f'éfl‘z’é‘? LIBRARY ' MiCl ita- State University This is to certify that the dissertation entitled DECAY OF CORRELATIONS FOR DYNAMICAL SYSTEMS WITH UNBOUNDED DISTORTION presented by CHARLES HOWARD MORGAN, JFi. has been accepted towards fulfillment of the requirements for the Ph.D. degree in Mathematics «(g Major Pr fessor’ Signature 7712‘, 0,3 Date MSU is an Affinnative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIRC/DateDuepes-sz DECAY OF CORRELATIONS FOR PIECEWISE SMOOTH MAPPINGS WITH UNBOUNDED DISTORTION By Charles Howard Morgan, Jr. A. D I S- SERTATI ON Submitted to Michigan State University in partial fulfillment. of the requirements for the degree of DOCTOR OF PHILOSOPHY D‘:}.)%F7tll"91‘tt “if B-iathematics \J O.) ABSTRACT DECAY OF CORRELATIONS FOR PIECEWISE SMOOTH MAPPINGS WITH UNBOUNDED DISTORTION By Charles Howard Morgan, Jr. In [YouQS], Lai-Sang Young introduces a method for showing that a large class of dynamical systems have exponential decay of correlations for Holder continuous real— valued functions (observables). She assumes that cumulative distortion along orbits is uniformly bounded. We extend this result to include dynamical systems which have unbounded cumulative distortion along orbits. Furthermore, Young shows that piecewise hyperbolic systems with finitely many domains of invertibility fit into her theorem. We Show that Young’s proof can be extended to include systems with countably many domains of invertibility, provided that the domains of invertibility decay at least exponentially quickly in measure. ACKNOWLEDGMENTS Every thesis is a collaborative effort with significant contributions from scores of people. While my name appears on the front page of this work, my contribution to it is the least important; the Inagnanimous, selfless efforts of the many people I list here (and the many people I should have listed but for whose names there was not sufficient room) are the real reasons for the existence of this work. I would like to thank my advisor, Dr. Sheldon Newhouse, for his patience with my many questions and for his kind encouragement. He was always a valuable source of information, ideas, and critiques which continually drove my work forward. I must thank Dr. Bruce Ebanks, who guided me through my work on my bach- elor’s thesis, and Dr. Lee Larson; both encouraged me to pursue graduate studies in mathematics. Thanks are due also to Dr. Clifford Weil who taught me real and complex analysis and who always gave me helpful guidance throughout all my grad— uate studies. Thanks also to Dr. Michael Frazier and Dr. William Sledd for their many helpful discussions throughout the preparation of this thesis. Many thanks to Prof. Gaye Holman whose guidance early in my college career helped me to start the path which led here. Other people in the Department who have had active roles in my success are Ms. Barbara Miller, Dr. Susan Schuur, Dr. Wellington Ow, Dr. John McCarthy, Dr. William Brown, Dr. Jay Kurtz, and Dr. Wei-Eihn Kuan. I really should list every faculty member and graduate student in the Department here, but there just is not sufficient room. iii I also owe many thanks to Dr. George E. Leroi, Dean of the College of Natural Science, and Dr. Doug Estry, Associate Dean, for their support and encouragment and the privilege of working with them on the Dean’s Student Advisory Council and giving me a very different viewpoint of the daily operation of a university and the difficulties which accompany it. My deepest gratitude goes to three people without whom I could not have accom- plished anything. I thank my parents, Charles and Lee, for all that they did so that I could pursue my degree. They sacrificed much for me. I cannot forget the infinite patience my wife Iffa showed me during the many days when my research wasn’t quite so promising. I never would have finished this work without Iifa’s support. New I hope that I can be as much support to her as she finishes her thesis. iv CONTENTS 1 Physically observable measures and correlation functions 1.1 SR8 measures ............................... 1.2 Correlation functions ........................... 1.3 Invariant measures and the transfer operator .............. 1.4 An introduction to the tower method .................. 2 The Main Theorem 2.1 Setting and assumptions ......................... 2.2 Statements of main results ........................ 2.3 Differences between Young’s result and ours .............. 2.4 Proof of Theorem 2.2 ........................... 2.5 Proof of Theorem 2.3 ........................... 3 One-dimensional examples 3.1 Two domains of invertibility: a motivational example ......... 3.2 F initely many domains of invertibility .................. 13 14 18 19 23 26 52 53 56 3.3 Countably many domains of invertibility ................ 57 4 Newhouse—Jakobson maps 59 Bibliography 66 vi CHAPTER ONE Physically observable measures and correlation functions 1 . 1 SRB measures In his 1976 paper [Rue76], David Ruelle extended the work of Sinai [Si1168] by intro— ducing a new type of measure for Axiom A attractors. This measure is invariant1 with respect to the given map, and it maximizes a quantity related to Pesin’s For- mula,2 and it governs the behavior of a set of orbits of positive Lebesgue measure. This measure ,u is supported on the attractor E, and for some positive Lebesgue mea- 1 71—] sure set of points in some neighborhood of Z, we have lim — Ecflf'x) —> / cpdu n—+oo n i-O for all continuous real- or complex-valued functions tp. We say that such a measure is physically observable and call it a physical measure, and we call the functions such as 99 observables. lA Bore] measure it is said to be g-z'nvan'ant if, for every Borel set E, p(g‘1E) = ;1(E). 2We discuss Pesin’s Formula in the next two pages. Throughout this thesis, we shall use A! to denote a smooth compact finite- dimensional Riemannian manifold (possibly with boundary). Suppose that f : ll! —+ A! is C2. Let A1 > A2 > > A, be the distinct Lyapunov exponents of (f, ,u), and let E1, E2, . . . , E, be the corresponding eigenspaces, respectively. In the same paper, Ruelle also showed that for Axiom A attractors u is the unique f-invariant measure which maximizes the quantity h,,( f) — f 217:, A, dim E,- du, where this sum is taken over all Lyapunov exponents greater than 1 and where h.,,( f ) is the metric entropy of f with respect to u and that this maximum is zero. In other words, this measure u satisfies Pesin’s Formula (1.1) h,,,(f) = [Z A,- dim E,- (1“. i=1 In 1968, Sinai [Sin68] showed that measures satisfying (1.1) for Anosov systems had the interesting property that their conditional measures on unstable manifolds are equivalent to Riemannian measure on those manifolds. In [Rue76], Ruelle proved the same for Axiom A attractors. Later, Ledrappier, Strelcyn, and Young (both independently and collectively in [Led84], [L882], and [LY85]) showed that, assuming (f, a) has a positive Lyapunov exponenet u—a.e., Pesin’s Formula holds if and only if the conditional measures of u on unstable manifolds are absolutely continuous with respect to Lebesgue measure. This leads us then to a definition of SRB measure which is as follows. Definition 1.1 Let f, 1V], 8, and u be as above. An f -inva7iant Borel probability measure it is called an SRB measure for f if it is ergodic, if f has a positive Lyapunov exponent u-a.e., and if the conditional measures of ,u on unstable manifolds 2 are absolutely continuous with respect to Lebesgue measure. We should perhaps mention that the uniqueness of the SRB measure for Axiom A systems as discussed by Ruelle has nothing to do with Pesin’s Formula. Instead, uniqueness is a consequence of the fact that Axiom A systems are mixing3 with respect to their SRB measures. It is possible for some dynamical systems to have many SRB measures, and we discuss such a system in the following paragraph. It is in general not an easy task to decide whether a dynamical system has an SRB measure or whether it is unique when it does have one. In [Ryc83], Rychlik con— structs a piecewise expanding interval map f : [0, 1] —* [0, 1] with countably infinitely many domains of invertibility which has finitely many SRB measures; however, Hu and Young in [HY95] construct examples of dynamical systems which are hyperbolic everywhere except at a single point and which do not have SRB measures at all. In the examples we consider here, all SRB measures for piecewise expanding maps are absolutely continuous;4 however, Rychlik also considers a map f : [0,1] ——> [0,1] for which the Dirac measure 60 supported at zero is an asymptotic measure. This map is defined as follows: (*) f(0) = 0 and f(:1:) = 2:1: — 2’].+1 if a: E (2"j,2_j+1] for all j E N. Rychlik’s example has two fundamental difficulties which our examples do not have, and so we avoid the possibility of having only singular asymptotic measures (for one- 3A dynamical system (f, p) is said to be mixing if, for every borel sets A and B we have “mu—+00 fl (f‘"A D B) = #(A)H(B)- 4Our dynamical systems live on Riemannian manifolds, so when we say that a measure is ab- solutely continuous, we mean of course that it is absolutely continuous with respect to Lebesgue measure. dimensional maps at least). The first of these problems is that f has countably many invariant measures which are indistinguishable from 60 by some regularity properties (absolute continuity, for example); in fact, 62-x is an f—invariant measure for every k E N U {0}; and the second of these problems is that f is not topologically mixing. We say that the system f : M ——> M is topologically mixing if, for any two nonempty open sets U and V in M, there is a positive integer N = N(U,V) such that, for every n > N, f"(U) F) V aé 0. For Rychlik’s map (*), the point 1 is a global repellor of the system, and 0 is a global attractor; thus, we can clearly find U and V in the unit interval so that f "U and V never intersect. 1.2 Correlation functions An important statistical property related to the SRB measure is the rate at which two random variables become increasingly more independent. Let f : M ——> M be a map which has a unique SRB measure ,u, and let «p : M ——-> (C and ib : M —i (C be random variables (observables). We say that go and t!) are independent if /Wdu=/ 0 such that ]/(990fn)1l1dI/—/cpdu/u’idu < C(nt’th" for all n 2 0. Similarly, we say that (f, 12) has polynomial decay of correlations if there is p > 1 such that < C(wlin'” [feoofnwdu—fsodu/wdu for all n 2 0. In particular, we are interested in knowing the rate at which the correlation func- tions C'Wp(n) tend to zero as n —> oo; i.e., when (,9 and it become increasingly more independent. One goal of our work is to find some rapid rate of decay of these correla- tion functions. One reason for wishing to know how quickly they decay to zero is that this rate of decay is an important factor in determining whether the Central Limit Theorem holds for some class of random variables. One rule of thumb is that, if the 5 sum ZnEN C¢,,),(n) is finite for all random variables in some class, then the Central Limit Theorem holds for the subclass of those random variables which are in L2(u). (See [BalOl].) It is perhaps easier to think of the rate of decay of the correlation functions as the speed at which mixing occurs. Recall that a dynamical system (f, u) is said to be mixing if lim,H00 u( f ‘"A F] B) = p(A)u(B). If the space of observables in question contains the characteristic functions of the Borel sets, then we have that Guam) = u(f‘"A n B) - HUD/AB)- Therefore, it is sometimes said that. a system (f, u) is mixing if C¢,ut(n) —+ 0 as n —> 00 for all and E L201). While different speeds of mixing have been considered in dynamical systems, dy— namical systems with exponential decay of correlations have received the most atten- tion, in part because many elegant systems exhibit this speed of mixing; however, the real reason they have gotten the most attention is that the two most well-understood methods for determining the rate of decay of correlations (contraction in a Hilbert metric on a lattice and spectral gap of a transfer operator) are designed precisely to determine whether or not speed of mixing is exponential. These methods do not detect slower rates of decay of correlations. Systems with true polynomial decay of correlations (i.e., polynomial upper and lower bounds on correlation functions) are relatively new. (See, for example, [Hu].) Not much is known in general about these systems. To this author’s knowledge, all dynamical systems with true polynomial decay of correlations have one of a few very specific limited forms. We consider here a class of dynamical systems which have exponential decay of correlations for functions which are piecewise Holder continuous, and we do this by looking at recurrence times of a subset of the phase space. Young has recently shown in [You99] that the rate at which certain subsets of the phase space return to another given subset determines the rate of decay of correlations; therefore, a slow rate of return of these subsets causes a slow rate of mixing. It is hoped that the extension we have been able to afford here to the work in [You98] will also work essentially unchanged in [You99] so that we can add to Young’s library of dynamical systems which have polynomial decay of correlations. 1.3 Invariant measures and the transfer operator The primary tool for proving exponential decay of correlations is the Perron—Frobenius operator; in fact, its spectrum is the key point of our discussion. Several of its properties make this operator a natural object for study. Let ,u be a finite Borel measure on (M, B), where B is the Borel o-algebra and M is as already described. Let g : M ——> M be some smooth measurable function, and let (,0 : M ——) R (or C) be some real- or complex-valued observable. We define the Perron-Frobenius operator P = L101) -+ L‘Ut) by _ 90(16) (12) 1990(9) — 3.9%; —_|det Dg(a:)|' A few basic properties of P are as follows: (i) It; (1090de = fit: so (if) 0 g) du for all cat/1 6 L100; 7 (iii) if V = (150 u is an absolutely continuous g-invariant measure, then P¢o :2 (to; and (iv) if P6150 = €150, then V 2 do ,u is an absolutely continuous g-invariant measure. All of these properties are immediate from the definition of P. We refer the reader to [Br096] for excellent, concise proofs of all four properties. We note also that (i) alone may be taken as the definition of P, and then the remaining four properties and the explicit formula for P follow immediately. 1.4 An introduction to the tower method In this paper, we use a construction which has only recently become fashionable in proving exponential decay of correlations. Instead of working directly with our dynamical system f : M —+ M, where A! is some finite-dimensional compact Rie— mannian manifold and f is some piecewise smooth measurable map, we construct a tower sitting over some prudently chosen subset A of [V1, and we use information about when points from A return to A under iterates of f. We discuss here how to do this in a more general setting. Let (X, B, ,u) be some probability space, and suppose that A C X is a measurable set such that u(A) > 0. We do not exclude the possibility that A = X, as one could do when considering Anosov diffeomorphisms. Suppose also that f : X —> X preserves u; i.e., u(f‘1E) = u(E) for all E E B. For each x E A, let R(x) E N be the smallest positive integer such that f R(x)($) E A. We call R(x) the return time of x. In [Shi96] one finds the following theorem (and proof) about returns of points in A to A. Theorem 1.3 Let f, (X,B,,u), andA be as above. Then R(x) < 00 for u-a.e. x E A. Proof: For each n E N, let us define An = {x E A : R(x) = n}; i.e., An is the set of all points which return to A at time n. Let A0 = {x E A : f"(x) E A for all n E N} denote the set of points which never return to A. It is clear that all of the Ans are pairwise disjoint, and it is Clear that A = U An. n20 Since f is measurable, all the Ans are measurable since A1=Aflf‘1A n—l A, : A nf-"A n fl f“(M\A) for n 2 2, and i=1 A0=A\UA,,. nEN Because A0 C A, no points in A0 can return to A0; in other words, if x E A0, then f 1(x) «at A0 for all i E N, otherwise we would have a return of x to A. This implies that f _i{x} (1 A0 = (ll for all i E N whenever x E A0 as well. Thus, the collection of sets {f’iAo : i E N} is pairwise disjoint. Thus, ,u (UieN f’iAo) = 22.61,, u (f‘on). Since this sum must converge to something less than 1 and since f preserves the measure u, it is immediate that MAO) = O, which is what we wished to prove. I This theorem permits us to partition A, modulo a set of u~measure zero, into the collection of Ans. For i 76 3' it is clear that all the images of A,- are disjoint from all the images of A], except perhaps when they return to A. For those who are familiar with the use of towers in recent results on decay of correlations, we should note that in the current context we are not making any as- sumptions about how the Ans return to A; in particular, we are not assuming that they map onto A when they return. Imagine now that we make a tower by constructing a column above each An by placing above it a copy of f An and above that a copy of fQAn and so on until we reach fn‘lAn, which is the top level of the column since ann goes back to A, the bottom of the tower. Let us denote this tower by A. We coordinatize the tower by identifying each point with its preimage in A and the image of A,- in which it lies; i.e., if y E ka, for 0 S k < i, then let x = f‘k(y), and identify y with the point (x,k). We say that (x, k) is on the kth level of A. We may define the tower explicitly as A={(x,i) : xEAandOSi M induce a map f : A —> A which moves points up the tower, except for points which are already at the top, and f moves these points to the bottom of the tower. The definition of f is as follows: - . (x,i+1) if R(x)>i+1 f (:13, 2) = (lex)(x),0) if R(x) = i +1 We refer to f as the tower map, and we refer to the map f R : A —> A, defined by fR(x) = fR(x)(x) as the return map or (in the language of Kakutani [Kak43]) as the induced map. 10 The measure u on .M also induces a probability measure it on A. Ergodic prop- erties of the induced system (f R, it) and (f, ,u) are proved in [Kak43] and in [Kac47]. Some of these results are summarized very nicely in [Shi96, pp. 23H]. The above results are purely measure-theoretic, but because we are dealing in this paper with differentiable dynamical systems and because our primary goal involves mixing, we must add some additional conditions to the definition of return time to guarantee that when the Ans return to A they satisfy some geometric conditions which are essential in our determination of the speed of mixing. We will have the additional problem that two points which return at the same time to A may not be comparable to each other because they land on opposite sides of some discontinuity set before they return to A, and so they will have to be placed in different A,s which return at the same time. Hence, the tower we consider in Chapter 2 will be more complicated than the above tower. It may have countably many columns of the same height, and this does present some complications which are not present in the above tower and which are not present in [You98]. We will discuss in Chapter 2 the precise definition of return time we require for our purposes. Since the publication of Lai—Sang Young’s paper [You98], the tower method ap- proach to proving exponential decay of correlations has become a relatively widely- used method. We claim that this justifies our attempt to extend her result so that it applies to a wider class of dynamical systems. In her paper, Young showed how her method applies to piecewise hyperbolic (with finitely many pieces) systems in two dimensions. Later Chernov in [Che99] showed how her method can also be applied to piecewise hyperbolic systems in higher dimensions. Alvez, Luzzatto, and Pinheiro in 11 [ALP] use the tower method to show that the rate of growth of Lyapunov exponents determines the rate of decay of the return times. These are just a few examples of uses of the tower method in this fashion; a multitude of authors are using the tower method in some way to determine rate of decay of correlations for various dynamical systems. Some authors such as Young in [You99] and Buzzi and Maume—Deschamps in [BMD] consider only tower systems instead of beginning with a dynamical system and constructing its tower. This approach usually assumes a priori the existence of a bounded SRB measure for the tower system and the result of Lemma 2.4 (iii) on p. 29; however, we shall prove Lemma 2.4 from a few assumptions on our dynamical system; we state these assumptions in the next chapter. 12 CHAPTER TWO The Main Theorem In this chapter, we prove two primary things: 1. dynamical systems satisfying certain conditions have SRB measures, and 2. dynamical systems with SRB measures which satisfy certain other properties have unique SRB measures and exponential decay of correlations for functions which are piecewise Holder continuous. We follow Young’s argument very closely; however, we make two changes which allow us to consider piecewise hyperbolic maps with countably many domains of invertibility instead of only finitely many and maps for which certain types of distortion estimates grow quite quickly instead of remaining uniformly bounded. 13 2.1 Setting and assumptions We assume that Al is a smooth compact finite-(1imensional Riemannian manifold, and that f ‘: [M -—> A! is a map. In all that follows, we let m denote Riemannian measure on M. If 7 C A! is a submanifold, then we use m, to denote the Riemannian measure on 7 induced by the restriction of the Riemannian structure to 7. Definition 2.1 Let H C M. We say that H has a hyperbolic product structure if there are a continuous family of stable disks F3 = {73} and a continuous family of unstable disks F" = {7"} such that the following are true: (a) for each 78 E F3 and each 7" E F“, dim 73 + dim 7" = dim AI; (b) each 7S disk is transverse to each 7" disk, and the angles between them are uni- formly bounded away from zero; (c) each 73 disk intersects each 7“ disk in one point; and (d) H = U 7“ 0 U 78 71‘ E F" 73 E F3 In other words, H is a like a rectangle which is coordinatized by two transverse foliations. We call FS and F" the defining sets of H. We shall call H' C H an s- subset if it also has a hyperbolic product structure and if its defining sets are P“ and some proper subset of F3. We define u-subsets similarly. We assume that we have some set A C A1 with a hyperbolic product structure, and we do not exclude the possibility that A = M. Furthermore, we require that m,(7 0 A) > O for all 7 E F“. 14 We assume also that A is decomposed into countably many pairwise disjoint s— subsets A1, A2, . . . such that the following are true: (a) for each 7" E F", my; ((A \ UieN A,-) H 7") = 0; (b) for each i E N, there is some minimal R, E N such that f R‘A, is a u-subset of A such that, for all x E A,, f”“ (73(x)) C 73 (fR'(x)) and fR" (7“(x)) D 7“ (f R" (15)); (c) there is some R0 > 1 such that R, 2 R0 for all i E N; and (d) for each i E N and for each 7 E F", m, (7 O A,) > 0; and (e) for each i E N, fR‘ lAi is injective. We should note that in the general situation, given a collection of Ais, there is a first return time for each A,, and it is possible that the first return time could be 1, as it is in the examples we consider in later chapters. To make the first return time larger than some arbitrary R0, we simply run the system forward R0 iterates, and then pull back A by each f IAi for R0 times so that we divide each A, more finely. The tower map and Markov partition We obtain our results on decay of correlations by working with a new map for which our current system is a factor system. Let U"EN f"A denote the union of all the images of A. We shall construct a tower A and the induced map F : A —> A as we did in Section 1.4 such that there is a projection 7r : A ——+ U f"A with the property nEN that f 0 it = 7r 0 F. For simplicity, we shall denote by R : A —> N the return time function RlA, E R, and by f3 : A ——+ A the return map defined by fR(x) = fR'(x) if 113 6 Ai- Let us define the set A as follows: A: {(x,l) : xEAandl=O,1,...,R(x)—1}. We shall define F : A —> A by (:c,l+1) if l1} x {t}. Then each A, is a canonical copy of the union of A,s which return after time t, and we imagine this copy of A, as sitting over the copy of A, in A0. The map F simply either carries the copy of A, on A, bijectively up to the copy of A, on A,“ if R, > Z + 1 or injectively onto a u-subset of A0 if R, = l -l- 1. The A,s induce a countable partition M, on each A, and, hence, a countable partition M on A for which the map F : A —> A is fully Markov. Throughout this paper, we shall use A,,, to denote the copy of A, on the lth level of the tower A, provided that R, > t; i.e., A1,, = A,flFl(A, X {0}), IfIf/i >1. A useful tool for defining a metric on the tower A is a function called the sepa- ration time function 3 : A x A —> N, which we shall define now. Let x, y E A, and 16 let M(x) and M (y) denote the element of M containing x and y, respectively. We may now define s as follows: s(x,y) = sup {n E N : M(ij) = M(ij) for all 0 _<_j S n}. In other words, the separation time of x and y is the last time that they are together is they travel through the elements of the partition M, or it is equal to 00. It is not necessary to say what s(x, y) should be if M(x) sé M(y) since all of our conditions are concerned only with two points which start in the same element of M anyway; however, one could for completeness simply set s(x,y) = 0 if M (x) # M(y). The separation time function 3 : A x A —> N induces a separation time function on A x A, and we will deliberately be sloppy and call them both 3. Now that we have established a separation time function, we may state the re- mainder of our major assumptions. In the following, let f“ denote the restriction of f to 7“ disks, and let D f“ denote its derivative, called the unstable derivative; we call det D f“ the unstable jacobian. We suppose that there exists a E (0,1) and that, for each i E N, there exists some C (i) > 0 such that: (A) for all x, y E A, such that y E 73(x), we have d (fnx, fny) S C(i)a" for all n 2 0; (B) for all x,y E A, such that y E 7"(x) and 0 g k S n < s(x,y), we have (a) d(f"x,f"y) S C(i)as(f‘y)‘", and deth"( (fix) 10g.1:Ideth“( (ny ) £00008 8(Iy)- n (C) for all x, y E A,, 17 det Df“(fjx) det Df"(f”J ) (a) for y E 73(x ), we have logH 3 C(7 )0" for all n > O, and (b) for 71,72 E F“ and 971.72 : 7, (i A, —-+ 72 (1 A,, the stable holonomy map, . . . d d’lm . we require that 071.72 18 absolutely continuous and that —————( * 72) = ('m71 00 deth“( (fjx) . Hdeth"( (fJ( (0x)) (D) there is a function g : N ——> R such that 9(a) —> O as n —+ co and, for all n E N 772., (7OUA,) S p n i>n (E) the sum 2 6N C( (i) “”90 ) converges. and for all 7 E I‘", 2.2 Statements of main results Theorem 2.2 (Existence of SRB measures) Suppose that f : M —> M with A C M satisfies the construction given in the previous parts of this chapter and that the following additional criteria are satisfied: (i) ZEN C(i)aR‘ converges; (ii) ZieN Ee0(i)g(i) < 00 for some 7 E F“. Then f has a finite SRB measure, which we call V. Let 0 < n < 1, and let 5,, be the space of observables defined by z {901 M —> (C 302(99) > 0 such that ]p(x )— go(y__)| < 6(9/ )d(:,1: y)" } 18 Theorem 2.3 (Decay of correlations) Suppose that f : M —> M with A C M satisfies the construction given in the previous parts of this chapter and that there are some Co > O and do < 1 such that m, {x E 7 O A : R(x) > I} < Coda, and (i) gcd{R,} = 1, or (ii) (f, V) is totally ergodic (i.e., (f",V) is ergodic for all n E N). Then (f, V) has exponential decay of correlations for functions in 55,, for all 0 < n < 1. 2.3 Differences between Young’s result and ours It is important here to list some of the ways in which our approach here is fundamen— tally different from that of Young in [You98]. Countable Markov partition The proof of the quasicompacity of the Perron-Frobenius operator in [You98] requires that the Markov partition M, on A, be finite for each l E N. To construct this partition, one requires a separation time function so, as we discuss below, which is different from the one we define above. This function so is used to define the Markov partition and to define a new separation time function s like the one we constructed above. By including the rate of decay of the A,s in our estimates of the spectrum of the Perron-Frobenius operator, we are able to allow for a countable partition M, on A, for each I E N. We are also able to take the natural partition of A into copies of 19 the A,s so that the partition is given by the behavior of the map f on A. An added advantage comes in the way we are able to define a universal separation time function, as we discuss next. Universal separation time The construction of a partition similar to our partition M as given in [You98] requires the a priori existence of some separation time function so : A X A -—> N satisfying the following three conditions: (a) so(x, y) 2 0 depends only on the 73—disks containing x and y; (b) for 33,3] E Air SOC/Bay) Z If, + 80 (fszafR‘y); (c) for x E A, and y E A,- with R,- = R,- and i #j, so(x,y) < R, — 1. This function is not necessarily unique for any given dynamical system, but it is in some way intrinsically constructed for a particular dynamical system. Note that a fourth condition in [You98] that only finitely many A,s are separated by time n for all n E N is not satisfied by our set of assumptions, but we show in this thesis that this requirement is no longer needed. Instead of requiring the a priori existence of some function 3 : A XA —> N satisfying the four conditions given above and then defining the partition M so that 3 may be extended to so to be compatible with it in terms of those four conditions, we define our partition M by the natural structure of A, and then we use M to define a return time function 3 which trivially satisfies the first three properties for any dynamical 20 system satisfying Condition (B). The fourth property of so cannot be satisfied since we may have countably many A,s returning at any given time, but we avoid this problem as we mentioned above. The disadvantage of our approach is that our Conditions (A)-(C) (See p. 17.) cannot be stated without first constructing the tower A; however, this is only an aesthetic problem, not a fundamental one since the tower is a canonical structure given by the hyperbolic product structure of A and the fact that f R is fully Markov. Non-uniform contraction Recall that C and a are fixed in [You98] and that, for each i E N, C (i) is fixed in our result. By Condition (P3) in [You98] for the case n = 1, we have (2-1) d(f$,fy) S Ca; however, by Condition (A) in our result we have that, for each i E N, (22) d(fm$,fmy) S C(ilam for all m E N. In other words, in [You98] all stable disks grow to a fixed size before contracting. In our result, for every m E N, there can be i E N such that C (i)am > 1 so that only finitely many stable disks need to contract by the mth iterate, but countably infinitely many may still be growing before their contraction begins. 21 Unbounded distortion By assumption (P4)(b) in [You98], the distortion along unstable leaves is uniformly bounded; i.e., for x,y E A, with y E 7"(x) and S = s(x,y). Then det Df“(fo) 0 (2'3) dethuUSy) S 8 ° 1 however, in our paper we allow that det Dfu(fs$) < 600') (2'4) deth“(fSi/) '— for all y E 7”(x) (1 A,. The only restrictions on the speed of growth of C(i) is that the sums ZR,eC(i)g(i), ZemilaR‘, and ZC(i)eC(i)g(i) must converge; thus, as iEN iEN iEN long as p(i) decays sufficiently quickly, C (i) can grow quite quickly as well. Hence, the distortion between two points is not necessarily uniformly bounded. Worsening of absolute continuity of stable foliation Let x, y E A,. Assumption (P5)(b) in [You98] requires that d(9:lm7’) S 60, dm, (2.5) but in our paper we allow the Radon-Nikodym derivatives to grow but require —1 (25) Ml S 800‘). dm, In particular, the Radon—Nikodym derivatives may grow without bound as i tends to infinity. 22 2.4 Proof of Theorem 2.2 We follow very closely the proof in [You98]; however, there are some significant differ- ences between our proof and that one because we allow countably many A,s to return at any given time, so we include the entire proof, but the biggest complications come because we allow C (i) to vary with i E N. Let 70 be some arbitrary 7u-disk which is full-width in some A and let uo 2: m , . We have already assumed that ’70 A70 1 . dllefflo (flax/40(7) dm, '70 (1 i0) uo(7o) > 0. For eachj E N and for each 7 E F“, let p} = whenever ( f R)Zuo(7) > 0; otherwise, let p} E 0. Suppose that 7 E F“ is such that (fR)i ,uo(7) > 0, and let x,y E 7 n A,. Let xo, yo E 7o F‘lA,0 be such that (fR)Jxo = x and (fR)jyo = y. Since x and y are both in A,, we know that s(xo, yo) 2 Rj. Suppose that xo, yo E 7o (1 Aio’ Then, by Condition (B)(b) for all 0 S k g Rioj -1 we have R- j—l (2’7) 1'1 det Dfu(fm$o) C , astrayo) — Iii-(n+1. det D fu( fmyo) ( 0) l/\ m=k By the Change of Variables Theorem and the Chain Rule we have, u R- j R- j-l pflx) : (18th (f 20 (150) 20 det Df"(fmxo) pf(y) det, Df" (fRiOJyO) m=0 det Dfu(fmy0) Ri j-l _ det D f"(x) 10—1 det D f"( fmxo) dethue) m2, deth"(f'"i/o) Bibi—1 d Df (f ) _ Ct u meo < C i - ( l ,1}, deth“(f’"yo) s C(i)C(iO)o,5(J?oa yo) — 12,0, + 1 < C(i)C(io)aS(“’y) 23 because s(xo, yo) Z R“, j + s(x, y). Recall that our separation time function 3 satisfies the condition s(x, y) 2 R, for all x, y E A,. We may then define . ' 8(any) ' Pa L:= sup eClzla Ssurpecula , . v . x,yzEE’iflA, 16 which is bounded since we assume that 2,61,, C(i)aR‘ converges. We have then that jet) 7(y) ‘b (2.8) S C(io) logL for all x, y E 7 F) A,, (P and this IS independent of j and 7. This implies that 1. In fact, AIo(i )= C(io) logLsupyemA p]. (y). We must show that lilo is also independent of i. Let x, y E 7 H A with x and y in different A,s, and let c : [0, 1] —> A be a smooth curve connecting x and y such that, for each i E N, we have c([0, 1]) fl cl E,, where cl denotes the closure, contains at most one connected component. For convenience, let us say that x is on the “left” of the curve c and that y is on the “right.” Let x, = aloft“ O A,). Then ()2: _p__,(:rz) iw HUT— iEN pj(x1+l) iEN : eZiEN C’(Z)CIR1 which is also finite. Therefore, there exists Mo > 0 independent of j, 7, and i such that (2.9) 1. The remainder of our proof is identical to Young’s proof; however, we include it here for completeness. Let w C 7 F] A, be open in 7 n A, with m,(0w) = 0, and let 8,, denote the s- subset of A whose section in 7 is w. Let U be a u-subset which is also a compact neighborhood of 7. Then (2.9) together with (C) imply that for all j we have 1 _mxs...) S (fRHMoWfl S...) < Arrows“) (2.10) . . _ 1111(2) m,(A) (fR){,,O(U) m,(A) for some AI,(i). The bounds in (2.10) also apply to Vo. By taking U arbitrarily small, the Martingale Convergence Theorem gives us that 1 .m,(Sw) 911(1) m,(A) 7n'7(Sw) 772.7(A) S Vd(5w) S 1111(1) ' for almost every 7. Since w is arbitrary, the density statement for V3 follows. We have so far constructed an f R-invariant finite Borel measure Vo on A with absolutely continuous conditional measures on 7“-leaves. We may clearly identify Vo with an F R-invariant measure i7o on Ao. We define a measure V on A by I; I: ZFBEOl{R>J}. yes The fact that V3 << #0 along with our assumption that m,(7 H A) 21.61,, R,eC(flg(i) < 00 for some 7 E I‘" implies that i7(A) is finite. We now define the measure V to be the push-forward measure induced 011 A by the canonical projection from A to A; i.e., ~ *V . . . . . V := %A_)’ where it : A ——> Un6N f"A is the prejection 7r(x,l) = fl(x). Clearly, V 18 V 25 f -invariant and the SRB property is clear since fflV clearly has absolutely continuous conditional measures on {fj7u} for every j E N, and these are unstable manifolds. This proves Theorem 2.2. 2.5 Proof of Theorem 2.3 A related expanding system It is not difficult to see that the Perron—Frobenius operator (1.2) improves the Holder continuity of observables when the map g : M -—+ [W is an expanding map and that it worsens the Holder continuity of observables when the map 9 is contracting. Because our dynamical system f : A —> A has both expansion and contraction coexisting, we must somehow “eliminate” the contracting direction. Expanding systems are mixing with respect to a measure equivalent to Lebesgue measure, but contracting systems clearly are not. Furthermore, expansion causes the Perron-Frobenius operator to improve certain properties of observables (e.g., Holder continuity or essential vari- ation), but contraction causes the operator to worsen these properties. There are two common methods for “factoring out” the contracting direction: (1) identifying points in the same stable leaf and considering the resulting quotient space, and (2) taking averages of observables along stable leaves with respect to some cone (in the lattice theory sense) of densities and considering the action of the Perron—Frobenius operator on these averages. The latter method requires a good deal of knowledge about the smoothness of the stable bundle (the collection of stable leaves); however, 26 the former method requires only absolute continuity of the stable foliation. We use the former method in our work, while Viana uses the latter method. (In particular, see pp. 79-122 of [Via].) We have a tower A and a map F : A —> A induced by the map f : A —+ A. We want to define a related dynamical system which is expanding so that the related Perron-Frobenius operator will reduce the norm of observables in the sense that ”in” S Allrpll + Klap], for some n E N, some A < 1, some K > 0 and for some appropriately chosen norm || - I] on some space of real-valued functions on A1. If x and y are in A, let x ~ y if and only if y E 73(x). Then ~ defines an equivalence relation on A, so A = A/ N is a quotient space. Let F : A —+ A be the map defined by F (f) 2 FE, where E is the equivalence class of the points in 73(x). Because F maps 73 disks to 73 disks, it is clear that, if f 2 7y", F(f) 2 fly); therefore, F is well—defined. In all that follows, we shall let A, A“ A“, and 77?- have the obvious meanings as given by the equivalence relation ~. We show here that for many dynamical systems there is a measure u on A whose conditional measures 011 7“ disks are preserved by the stable holonomy map; i.e., if 0%,: is the stable holonomy map taking points from 7 to 7’ by sliding them along the stable foliation, then 0w, = [1,]. This fact allows us to “collapse” A to A as described above in a way that. ensures that the essential properties of the dynamical system F : A ——> A are carried to F : A ——> A. We shall define a reference measure I2 011 A and then extend it to all of A by letting = 7‘71 A: 1. We shall also use the name fl when talking about this measure on Ill—51 A. This measure R will be such that JF E 1 on the points in A which are not on 27 the top level of the tower, but JF = fl? 0 F_(R_l) at the points which are on the top level of the tower. Let 7 E F" be some fixed 7“ disk. For each x E A, lot if denote the single point of Ain73(x )fl7. ForeachnE,N let deth( fix) do thu( fif)' By our assumption (C)(a) (See p. 17.) on each A, the sequence of u,,s converges uniformly to some function ulA,‘ On each 7 E F“, we define u, to be the measure a, = e“ - 1mm 771,. By Conditions (D) and (E), the measure u, is a finite measure for each 7 E F“. We note that f 3'“ (1 At is nonsingular with respect to the measures u, If we have fR' (70A,) C 7’, then we shall write J(fR)(x) for Jllmflv' (fR‘IVnAi) (x), d(T:1m2) d7". 1 where Jm,,m2(T) z: , since it will be clear in our estimates in which A, a particular x lies. Lastly, we note that iEN S ZR,eC m,(7flA) zEN S m,(7 O A) Z R,eC(i)g(i), iEN and this sum is finite by assumption (ii) of Theorem 2.2; therefore, fl is a finite measure on A. We prove next a lemma which shows that this measure fl allows us to collapse A along the stable foliation F3 in a way that preserves J ( f R). 28 Lemma 2.4 Let 7 and 7’ be two unstable disks in F“, and let 6 = 6,7, : 70A ~—> 7'flA be the stable holonomy map discussed earlier. Then the following are true: (i) 9,117 = M; (ii) J(f”)($) = J(fR)(?/) for all y E 73(18); and (iii) for each i E N, there exists C,(i) such that, for all x and y in 7 n A,, < C1(,)as(IRI,IRy)/2. ————1 |J(fR)($) J(f”)(y) The proofs of (i) and (ii) are exactly the same as in [Y01198]. For (iii) we follow this proof but take C,(i) 2 5C (i) The proofs of all three parts are very short, and we refer the reader to [Y01198] for them. A space of observables for the factor system We shall define a space of real—valued functions E : A —> R. We use the bar notation for the function names as well since we will introduce later a related class of functions (,0 : M —+ R. It is this latter class of functions in which we are interested; however, we will demonstrate decay of correlations for the factored expanding system with respect to the former class of observables and then show how the correlation functions of the original system are related to the correlation functions of the factored expanding system. Throughout the rest of this paper, we shall let IE], and Ifloo denote the L1 and L°° norms of E, respectively, with respect to the reference measure 17. By “951,, we shall mean ’g'o'I—A— , where AL, has the same meaning as earlier. We choose 6 > 0 such that 1.1 29 e‘do < 1. (See Theorem 2.3, p. 19, for the definition of do.) For future reference, we note here also that there is some K > 0 such that ”(130)211W )e“ A has an invariant absolutely continuous probability measure V : pp, where ‘p‘ satisfies cg] S p S co on AL, for some co > 1. Furthermore, [73(5) ——‘p(‘y‘)| S C’(i)fis(f*m for some C’(i). 31 Proof: We take Young’s proof as it is, but we make the obvious changes required 71-1 1 _. b our use of C i . For each n E N, let V” = -— FJ (" — ). Consider V" —- , and y ( ) n 14:; * ”Jle 8 A0 _ dfin _ 1 _. _- ._ . —-k _ let pn = 5. Then '0"le = R 2,031, where p}, is the density of F, (ulaj), and the 36 03s are the components of F ‘on fl Ao for i S n. Let x,y E Ao,, the ith rectangle —i on Ao, and let E'j' E of be such that Fif' = T and F ,7 = y. Then where i, < 12 < - .. < 2,, = i are the times when F 0] C Ao. Furthermore, we have that J? (EH?) J’F‘ (EVE) 5 exp (01(2‘1/33(f“5"_”"y’)) 3 exp (Clev‘is‘i'w) - Thus, we have that ,6}, ('y’) S 727,, (T) exp (Cfiiflislffl); hence, it follows that p" (y) S 7),, (T) exp (C1(i)fisl'_*§l). Letting n —> 00, we have p (y) S p(f) exp (C1(i)[33(f’yl) for all ff, Eq— 6 ADJ. Following the proof of Theorem 2.3, we know that the sequence {fin} has an nEN accumulation point, which we call V on A with 0 < V (A) < 00 because of our assumption that ZEN R,eC(i)g(i) is finite. Also by the proof of Theorem 2.3, we have some co > 0 such that cgl S p S co 011 A. Thus, we have that bl — —— j - sir—,— lfifi) -’fi(y)| S lpIZX—ul -]—[—y-]— —1]:£ some“ ‘y’ El whenever ify‘ E A,,,. I 32 Decay of correlations for the factor system _— The Perron-Frobenius operator P associated with the dynamical system F : A —> A with the reference measure fl is defined as usual by In order to prove that the factor system (F, F) has exponential decay of correlations for functions in f, we have to demonstrate three properties of the Perron-Frobenius operator, to which we shall simply refer hereafter as the transfer operator. We must show that o it is a bounded operator with its spectrum is contained entirely within the unit disk; 0 it can be approximated by a compact operator from f to f; i.e., there is some compact operator Q : .7: —> .7: such that ”PN — Q” < AN for some N E N and some A < 1; and 0 its only spectral point of modulus 1 is 1 and the corresponding eigenspace is one-dimensional. The first two properties are proved by showing that P satisfies what is commonly referred to as a Lasota-Yorke inequality: _ I _ I _ (2-13) IIPNwH S A” He?” + K hell, for some N E N, some /\ < 1, and some constant K > 0, all of which are independent of a This is precisely the inequality found in [IM50] and used there to prove the first 33 two properties as well. The third property is a consequence of our assumption that (f, u) is totally ergodic or gcd{R(;r) : :1: E A} = 1. In other words, the third propery will be true if the system (f, V) is mixing. Having derived (2.13), one could conceivably use the classical result of Ionescu Tulcea and Marinescu [IMSO] to prove the first two desired properties; however, this author has found it a more difficult approach than the one in which one approximates P by a compact operator. Recall that the function F : E ——+ K is one—to—one on the parts of E which move up the tower under F, but it certainly is not one-to—one on the parts of K which get mapped onto 30. Also, we have assumed that there is some N E N such that R,- 2 N for all 2' E N. This means that the first N levels of the tower Z are complete copies of A; i.e., they all contain all copies of the Ais; therefore, as we try to prove (2.13), we will have to consider how P behaves on the first N levels of A— separately from how it behaves on the levels above N. To prove (2.13), we shall require four estimates; two estimates for each of the norms II ' ”h and H ' ”00 on each of these two pieces of 3. Estimate 1 For alll 2 N and for every a e f, ”(19%),,” = e-CN ||¢,_N,,||oo. . . —N . ——N— Proof: Fix 1 and 2. Because I Z N, F 18 one-to—one on F AM, so we have “(131W = “PA/While.) e—"d(z') 00 = ess sup |¢(§)| e_(l”N)‘d(i) e‘N‘ 17677—sz _ 7 —N( — llama-Ilse , which is what we wanted to show. I 34 Estimate 2 For alll with 0 S l < N and for all §5 E 7:, we have Cl“) ”(PAH—’5) 2'” S _e_ eflkdfi) 9—5'1—-N— + 1‘ 0° ;H(Al,i) F Ali 1 K C1(i) N —16 —.1_— _ . 6 B 6 Q(’) W F NAM h Proof: Fix 1 and 2'. Then we have 1 2.14 H P”? I < —-—--1__ _ . .—-1__ _ "‘d ', ( ) ( 9’)” 00 _ ; I‘I—FN F NA” 00 ’99 F NA” 006 (Z) . ——N where Z,” means that we sum over all the inverse branches of F . We note now that 7.1__ __ “V F ”AL,- 1 jL _ _<_ _ _ $06177 + .___ .2. N 00 — F "A i F AJ (2.15) ”( l’) I ess sup '35 @1) ‘ @(lbll- 51326F—~ZM By (2.15) we decompose (2.14) into the sum of two parts: 1 1 S —_—°1———N— l ' __ _ ‘/___. _ Ed—M 346d“) ”(10%),.- ———-1——~— esssup My )“WWI e‘l‘dfil- ——N . __ _ l 2 JF F Al" 0° ylvy‘ZEF NAM For convenience, let us call the former sum (Suml) and the latter sum (Sum2). Then the distortion estimate 1 (2-16) T ° 1——N— _ e _ — JFN F Am 00 MAM) yields (8 1) Z 601“) 1 ‘d() um S _ _ E- __N_ e-6 2'. b, #(Am') F I: 1 . 1 2 01(2) ,—-,1__ _ _z._d. a e *9 F NA“- 8 [(2') (2) v1 |/\ |/\ Cl“) — —f€ ' e '1__ __ e 2 . EM 90 F N A1,,- 9( ) 1 . . . —-N-— To estimate (Sum2), let us use lb, to denote the level of the tower in Wthh F A1,,- in the branch in question, and let 3;, = ZR,” where 3;,” is the element of the top level of the tower through which the branch in question passes. Then the distortion estimate _ Z ) I—IT’ 1—-N— S 8010)“ (—br *- JFI F Al] 00 fi(Al,i) yields (Sum2) 1 _ 7 —_ 7 S 2 _—_N 1 —N I GSSSUP l‘p(y13(y 3:0)(y2)|e—lbre elbrf BRIG—led“) b, JF F Al: 00 EfflzEEF—NEIJ (3 1 2 . 1 _* . ,..n,,=esssup 2: ”(F “l -— “05” “l ELM/3,, b, J?” (F‘N 351) J? 154252) (2.17) . fi-S(F‘N§1,—F_N§2) e—l€d(’l) _<_: esssup ( ._ _ —-N— am) _ W172) JFWI) .1?”sz B—S(§1.372)) ,BNd(i). For each of the inverse branches we have that WE) _ 90(y2) ff” ('91) WW2) <1wa W2)! __ g < ml) 45%)! _ + 1F” (a) ._ __ —'—(lbr-N) _ —“‘(lbr_N) __ , s F '1 ,F tiff/93: 01W ( o.) (99)), JP (92) where l is as before the level on which 3;, lies in its branch. Combining this with (2.17) we have the following: <2 esssup ((I¢@Q;g(y2)l+ 57‘ 311 #261? NAM JF (yl) IESgQN Cl(i)fls(f—(zbr—N)(yl)f—ubr—Nuyfl)) H—s(y,,yz))fiNd(z-) ||< W”) JF (T19) S g 7%? 1F_NAII 00 El _3::S;11PAI I? ¢(yfl12(y—ly:)(y2ll fiNd(Z-) +;CI(9);|J;N F N A. '.,9. 1F_N_A_Hoo5~d(9). Note that we used the fact that 93 (TM—MT,FWMW) S 9491,99) because 8 (F_(lbr_N)yl,f—(lbr—NEQD Z s(y,, yg). We have already seen both of these sums in Estimate 2, and so we simply now follow our work there from which we get _ 01(0601“) _ PNLP .II S ———_———8 lcd(l) 90' 1—_N— ll( )1" h Z»: fl(Al,,-) F Au] KC ' (31(1') N 1__ __ + Me Be em a F NAM h +KeCI“)/3Ng(z') ,9 1—_N— Am h and this is the inequality we sought. I Combination of Estimates 1 through 4. We now derive our Lasota-Yorke in- equality (2.13). Combining the above estimates and using the definitions of ”$1,;- II we 38 have ”PA/9’3“ < 8—1“ ”9/3”“, + KBN (28) Cl“ 9(1)) llfllh iEN + [3Ne_N‘ ”:9”, + 21m” (201096 “"20 )) Ilffillh iEN 6‘7“"(10') 7 01(i )6 “Md“ ) 7 + (2]; 21(3”) ) l‘l’ll + (g; [_12(Al,i) ) lcf’ll s ((1 +fi“’)e “+31%” (201(2))9‘1“) 9(0)) IIZEH' iEN 01(2)eC‘(i)d(2) 7 + 2(1462 71- (31,1) ) lLFll g ((1 +,{3N)e N‘+3K;3N (201(2) 22‘1 ‘l 920)) II; II’ ieN + 2 (201(1) >600)? (2)) Iii/“I1 iEN Choosing N large enough so that it satisfies (2.12), we have some A < 1 so that ._ , . _ _ (2.13) ”PM; SANII¢II'+K'I¢|1 for some K’ > 0 since the sums in the last two lines are finite by Condition (E) on p. 18. This is the Lasota-Yorke inequality (2.13) which we sought to verify. Spectral radius of P. In order to make use of the approximation of P by a compact operator, a fact which we prove in the next section, we require that the spectrum of P is contained in the unit disk. By (2.13) we know that, for all k E N, ._ _ I _. -— PM 1)N¢)H SANHP(k-_)1)NSPIH +KI|P(k— 1)N50|1° It is a basic property of the Perron-Frobenius operator that |P¢|l = [Ell for any real- valued function a (See [Br096] for an excellent discussion of important properties of 39 P.) By this property and by induction on the above inequality we have k—l llPst‘éll’ s W H¢II’ + (WE/V”) lell i=0 for all k E N; therefore, we obtain “We = ||PkN¢||'+ |P’°N¢|l g (1+/\’°N + K’COZAJ'N) ”a“. jEN Thus, “PkNEH g [C ||E|| for all k E N, for some [C > 0, and for all E E 7-". Let n E N. Then there exists 2' E N such that 0 g i < N and such that n = kN + 2'; therefore, ||P"E|| S [C ”PE” 3 IC (supOSKN ||P2||) HE”. Since this is true for all n E N and for all E E f, the spectrum of P lies entirely within the closed unit disk. Approximation of P by a compact operator In this section, we shall construct a finite—rank operator and show that this operator is close to P in the sense used in [D858]; i.e., that there is To < 1 and m E N such that ||Pm — Q“ < Tm. Let M denote our original partition of E into the 31,,- components. Recall that each Z, is partitioned into countably many pieces. For k E N, let Pk be some finite collection of Ems for l S k such that 2 H (Km) 8’6 < 6k, where 6k -+ 0 3,,”th _ as k ——> 00. We can do this for the following reasons. The partition on each level A, of the tower Z is countable, and Ii (31) is finite. On the first [C levels of the tower, we include in Pk sufficiently many of the _A-Ms so that what remains has measure as small as we wish. For the levels above k, we know that their total measure is not more than C068. For convenience of notation, we will use Pk for both the finite collection and the union of its elements. Let E9C = E- 179k and E>k = E — E“. For E : Z ——> R, we shall 40 let E N (E) denote the conditional expectation of E with respect to the partition M and the reference measure fl. Let Qk : f —-+ .7: be defined by (2;, (E) = PN (EN (Esk)). Because the number of Ems in ”Pk is finite, Qk is clearly a finite—rank operator. Let E = (E — EN (E))Sk. Then EN (E) = EN (55' 1m) — EN (EN (315' 119;.» = 0. Note that (P” — Qt) (a) = P” (W + a”) — P” (EN ($90) = P” (as — EN W» + P” (wk) In order to estimate H (PN — Qk) (E)||, we must break this down into the four pieces corresponding to those in Estimates 1 through 4. In fact, we will encounter here nearly those very same estimates; however, these estimates will be easier since certain terms which appear in Estimates 1 through 4 will not be present here because E N (E) = 0. Estimate 5 Forl 2 N, we have ”(PW)“l S BNe—Nf l $1—NJHh' 00 . —N . —-N— . . Proof: Since I Z N, F 18 one—to—one on F A”, from the definition of H - “00 we get ”(WM : —. 1__ —-l(d ' lw F NAM 006 (z) I — — _ —, _ _6 . S ___N_ / wdfi + esssur) l1/J(:t/1)-w(y2)| 8 “1(1)- _. __N_ [1(F A”) r—"z and A1,. (,1' 41 By the definition of E, the integral above is zero. Furthermore, we also note that lw (3&1) — E (val = lab" (a) — EN (a) (a) — a (a) + EN (:5) (a)! : WW1) " $(y2)l because EN (E) (ijl) 2 EN (E) (ij2) since both 5, and fig are in the same branch of ——N . . . F and, therefore, in the same element of the partition Pk. Thus, we have N (PNELJ. g ess sup WW1) _ PWNI e—Iedm _ _ -——N— 311412617 Ala IOO , |¢(§i)_¢(lj2)l —(l—N)c —Ne N - _ _ _ ‘— . [33(ylvy2) 6 6 ’8 (1(2) -— —N N S llWi—N,i|lh€ ((3 , and this completes this estimate. I Estimate 6 Perl 2 N, we have (PNEhJHh 5 ”PI—Nallil/BN‘Z—M' Proof: From the definition of H - ||h we get _ PN— —- _ PN_ 7". ||(PN¢)I. = esssup |( 1/1)(:i:1)_ _( 2'1’) (“DH e—ied(z-) ,1 h 53152631,.- [841132) 2 esssup (111(1),):—j/J(g/2))€_(,_N),E e"N‘,8Nd(z') _ __ —_N_.. [33(y1iy2) yliy2EF A13 3 IIEI—NJHh’BNe—Nf’ which is what we wished to prove. I Estimate 7 For I < N, we have “(PNELHOO S KflNeC‘“)Q(i) E 1P_N—A_ . l,i h 42 Proof: All of the hard work for this estimate was done in Estimate 2, but here = 0. Therefore, 1 we have only the latter term from Estimate 2 since _.1__ _— IP F NAM the estimate we seek follows immediately. I Estimate 8 Forl < N, we have + h E1__N_ HeatsKeaneCWe-m» F Kemwgm 7.1__ ,_ 9” F “AL,- h Proof: Analogous to Estimate 7, this argument mimics the proof of Estimate 4 , . 1__ — ‘1’ F ”A... = 0. Thus, our estimate follows immediately. I 1 except that again Combination of Estimates 5 through 8 and estimate of IIPN (E>'°) II. Com- bining Estimates 5 through 8 and using the definitions of II ' ”00 and II - II h we have 7' ”PW” = HPWII' + IPWII = IIPWII' s 2/3Ne‘N‘ Halt. + KHN Hell. 2: 60““90) iEN + K3” Halt. Z 01(z')e01<”g(z‘)+ K6” Hall}. 2: e""“’e(i) ieN ‘EN 3 r3” (2 + 3K 2 Cl(i)eC‘“’e(i)) Hell- iEN Lastly, we show the most difficult calculation for IIPN (E>k)II, and the others are nearly identical. We have the following: 1 PN 55>}: I S I '1_—N— 7'1—-N— ([(i) '7 00 II( ( ))l,l 00 ; J—F—i'V F A” 00 5" F A” 00 IV’II >1: 1 Z J?" F ”A,.-I... .. F NA .0 MIMI... Ewen :sz 43 CiIi) III-(FA, Al i) eflbr — S :6 C1(z )# IV’ "00“ 37c NEH) + Z __1_W.1__N_ I¢.I__N_ dawn... E¢1> :lk + 66.,(.):(,)_@ Hall... 2 g (EM) 6:. [(i) s (eCMidufiig) II¢II' (at + ck), for some 6;: —> O as k ——> 00. Also 31363:: :19.- IPN(¢>k)II=I¢>kII= 2 I901; I <€Z|¢|1 31.4731; forsomeek—+0ask—>oo. We may then choose k E N and N E N sufficiently large so that 2:: (6k+ek+€Z) )Zeclh ) @22( )+fiN (2+3KZCIU (1")d( )) <1. iEN iEN Let To be such that Z < 76" < 1. Then II(PN — Qk) EII 3 “r6" IIEII. We now apply the following proposition from Dunford and Schwartz. (See pp. 709-711 IDS58I for the proof.) Proposition 2.6 If P is a bounded linear operator, if there is some compact operator Q, if there is some r0 < 1, and if there is some N E N such that IIPN — QII < ”r6", then any spectral point A such that I)\IN > IIPN — QII is isolated and its eigenspace is finite-dimensional. Thus, P is quasi-compact, and our only task now is to isolate 1 as the only spectral point on the unit circle. 44 Showing that (P,U) is mixing We have established that the Perron-Frobenius operator P : .7: —> .7: is quasi-compact; i.e., that all spectral points lie in the closed unit disk and that there is some disk of radius r < 1 such that all spectral points outside this small disk are isolated. When this happens, it is often said that P has a gap in its spectrum. In order to prove that (T5,?) is mixing, we must show that 1 is the only spectral point of modulus 1. Young proves in [You98] that (7,?) is exact, and her proof works here unchanged since this part of the proof does not rely upon the behavior of C(i); thus, we refer the reader there for the complete very brief proof. It is a basic fact that exact systems are mixing. Decay of correlations for (7,7)) It is worth noting that the mixing property of (PE) along with the quasicompacity of P implies directly that 1 is the only spectral point of P on the unit circle (and it is of course an eigenvalue since it is isolated) and its eigenspace is one-dimensional. To see this, let (15 E L1(fi) and 1,0 6 L°°(fi), and suppose that P45 = 0gb. Then 1930/11)(P"¢)dfi = "1330 (w of") (bdfi = lim (w o P”) 12—5- (ii? a... (5) Wu— =/¢(fi/¢dfi) am. 45 Thus, onq§ = P"E converges pointwise to E f Edfi, which implies that o = 1 and that the eigenspace of o is one-dimensional. We will show explicitly at the end of this chapter how this puts an exponentially decaying bound on the correlation functions of (FE) as well as for (F, V), which is the goal of our proof. Decay of correlations for the original system In this section we show that we may compare the correlation functions, which we derived in the previous sections for the factor system, to the correlation functions for the original dynamical system (f, V). \Ne have so far a Markov system F : (A, E) ——> (A,i7) over the dynamical system f : (M, V) —+ (ll/1,11) where V is an SRB measure. ~ 7r...V We note also that we have a projection 7r : A —> M such that V 2 ~ ( A have the factor system F : (A, ‘1?) —+ (3,?) and the projection if : A —+ A such that . We also v t if = i}. For E: [W ——> (C we let E := E 0 7r : A —> IR denote the lift of E to the tower A. Recall that 7r is simply an identification between the lth level of the tower and f‘A. Let 77 > 0 be as we previously defined it. We defined our space of observables 56,, earlier. For convenience of notation, let us use Dn(E, E; V) to denote the nth correlation function with respect to the measure V: Bdrm/AV) =/so(wof") dV—/EdV/Edu and the analogous definition of Dn (E, E; E), the nth correlation function with respect to the measure 3. Note that Dn(E, E; V) is with respect to the map f : [W —> A1, but 46 that Dn (E, E; D) is with respect to the map F : A —+ A. It is obvious by the Change of Variables Theorem that (2.18) Dun/mm) = D, (a 215; a) . It is our goal in the remainder of this chapter to show that Dn (E,E;§) can be approximated arbitrarily closely by quantities involving things related to the system (F,—A_, E) which we know already has exponential decay of correlations for functions in f. Because of (2.18), we shall show that Dn (E, E; I7) can be approximated well by the correlation functions of the system (F ,_A_, '17). We Show in the last two pages how the observables E : [W -—> IR in f)" are related to observables in f for the factor system (F,A,v). We state now a lemma which will be of much use later. We shall fix It later. In what follows, let A(i) denote the part of A sitting over A,. Lemma 2.7 Let :1: E A(i). Then diam (anM2k(:r)) g 2C(i)ok. Proof: Let y1,y2 E M2k($)flA(i). Then there exists g E 'y“(y1)fl'ys(y2). Suppose without loss of generality that M2k(.’E) fl A(i) C A,. Then nF‘ly), 7rF“’y2 6 A;, and they both lie in the same 73-leaf. By (P3) we have d(7rFky),7rFky2) S C(i)a’+k S C(i)ak. Similarly, nF‘lg}, nF‘lyl 6 A,, and they both lie in the same vu-leaf. By (P4)(a) we have d(7rFk§/,7rFky1) g C(i)aS(Fk9'Fky‘)’“+kl g C(mk. Thus, we have that d(7rFky1, ani/g) S 20(i)ozk. I 47 Estimate 9 Define E, on A by EkIA E inf{E(:1:) : :1: E FkA} for every A E Mgk. Then IBM (a, 27} o Fk; 5) — DH ((25, a; 27) I s C’( 0 Proof: Note that Dn—k (giJOFkiD)-D n— k(¢ iz/jkil/ V)I /E(E0FkoF"_k) dU—fEdU/EodeD —/E(EkoF"_k) dD+/Edi7/Ekdi7 +I/(E0Fk—i3kw’z7-fs‘o'dv s I/(EoFk—‘z/iquHw It follows from Lemma 2.7 that I f (2% W 4/3,.) elm—WU + wkI/(EOFk—EkM’J-fEdi} =Z/A( (EoFk_ — )c,—Fnk¢d~+ Z/A (EoF —¢k)d~/A 0 where C’( 0. Proof: Just as in the proof of Estimate 9, Lemma 2.7 gives us Z/A-( “(Wylie)“? W) MZ/A War/A 90- 9006117 iEN iEN akna where 77 is chosen as before and 02(90, 9’2)— — 2Bok77max|1/)|€()216NC(139) (z), and this sum is also obviously finite. 49 Note that in Estimates 9 and 10, we did not use the fact that 99 and 1/1 were Holder on M with a uniform Holder coefficient. Rather, we could have simply required that both 90 and 1,!) were Holder on each A,- with some Holder coefficient €(z', 90), for example, such that the sum ZieN {(i, 90)C'(z')g(z') is finite. In particular, this would allow €(i, (p) to grow without bound as i -—> 00. We now observe that Dn_k(§6k,;fik; 3) can be expressed in terms of objects only from f : (3,?) —> (3,7). First, / (a. o F“) :03. d3 = j a. d (Fr—k ($1.27)) = fEkd(F:l(¢ki;)) = [name w» = / W?" (W) dfi- Also, we observe that [ma/adv:farms»jam/adv-fadv. Estimation of the correlation functions and end of the proof We note that we have DH (nae) = If (a o F"-’~‘) adv— / sad;- fad; 5O S li/llem) '06, P" (W? — where H . H is the norm on f we introduced earlier. Next we observe that Dn(99,'9’1;1/) 2 0,, (975, 12;?) = Dn_k (93', E0 Fk; 3). We have proved earlier that P is q1.1asi-compact, so we may define T = sup{|(| :C E 0(P) such that C 751}. By Estimates 9 and 10 we have 13110709711); V) S (1912.019, #1; V) — Dn—k (Saki/:53” + Dn-k (amid?) 55k? “ (f—kpdfi) :5“. Now choose k z 721 and T1 = max{ak’7, 7'}. Then 7' is the rate of decay of the correlation 3 (CW: 2/2) + C"(<.0,1/1))01k" + lwlmmfla’w functions, and we define C(so, 1/2) = C'W, E) + Wet/0 + Whom—006' and the last factor is finite since it is clearly bounded above by ”W” + H (f Wdfi) 'p'“, and this completes the proof of Theorem 2.3. 51 CHAPTER THREE One—dimensional examples In this chapter, we construct some one-dimensional dynamical systems which do not fit into Young’s original construction but which fit into the setting of our extension of Young’s theorem. We begin with a map with two domains of invertibility and show how our technique extends trivially to a map with finitely many domains of invert- ibility as long as we choose A prudently. Finally, we show that under an additional assumption about the speed of decay of the measures of the domains of invertibility, we may extend our result to include maps with countably many domains of invertibil- ity. Some of the properties of our examples are not really essential for the dynamics we consider; however, our aim is to construct simple examples of dynamical systems which are not (as far as this author knows) covered by previously known theorems. We claim that it is clear which properties of our examples are essential and which are merely aesthetic. 52 3.1 Two domains of invertibility: a motivational example Let a 6 (0,1). We partition the unit interval [0,1] into two pieces [0,a) and [a, 1]. Furthermore, we partition the interval [a, 1] (mod 0) into subintervals by a countable collection {T_,- : iEN} such that a < < L,- < 'r_,-+1 < < 7-1 = 1 and such that [a, 1] 2 (Jim [T_,-_1,'r_,-]. To use the notation of Chapter 2, we let A,- = (T_,_1,T_,). For the moment, we wish to define our function f : [0, 1] —> [0, 1]. We let (i) inlays) = 3 Then for each 2' E N we suppose that f IA- is C2, and f satisfies the following condi— 1 tions: 1 ’ >—>1, flAil—o (ii) there is a < 1 such that, for each 2' E N, (iii) for each i E N, fA, = (ai,ai_l),1 f’(=r) f’(y) SK, (iv) there is K > 0 such that for each 2' E N we have sup x,yEAi (v) for each i E N there is D,- > 0 such that fill/M] :— D,-, and f”($) f’(a:) < K' (vi) for each i E N, —I We shall also assume that there is b < e‘1 such that m (U A.) _<_ b0 . We shall i>l see later that this will ensure that Condition (E) is satisfied. 1This property is not essential, but we use it here to make all the estimates easy. 53 To understand how this map fits into our scheme from Chapter 2, let us examine how it behaves on each A,. First, we let A = (a, 1). The subinterval A1 is mapped onto A in the first iterate, so its return time is R1 = 1. Then f maps the subinterval A,- onto (at, ai’l) and then onto (at-1, ai‘2) and so on until it is finally mapped onto A. The return time of A,- is R,- = 2', so we have the simplest possible example with one A,- returning at each time n E N. Let :13, y 6 A,, and let C(i) = K’a“i+1, where K' = szeN oz‘j. We claim that -—E%3]< iKZCFj < K’a _' = C(i)a_1. jEN First, let us note that, if :1: and y are far enough apart in A1, they will separate _(___fj$) f’ (f’ ) Condition (B)(b) for 0 S n < sa:( x,y y.) (See p. 17.) We should note that Conditions when they return to A, and so we need only look at the quantity in (A) and (C) are vacuous here since we do not have a contracting direction. For these :1: and y, we have _(___f':c M ———-§=| gm];— which is what we require. On the other hand, if :1: and y are quite close together in A,, they could return together to another A], in A. If they land close enough together in A3,, they could return together to another A 1,, and so on; however, because of the minimum expansion by a“, they must separate in finite time. Let us consider the case when Ia: — yl = m(A,), where m denotes ordinary Lebesgue measure. Because f “0, a) is linear, distortion is introduced only when fig; and fj'y are in A. If they land back in the same A,- in which they started, then they must now be far enough apart to separate when they return again, and so we would have f’(fj$) - —1 m] S (1+a)K _<_ C(z)a In fact, the only way that :1: and y can avoid separating when they return to A is if the land together in some AJ- with j < 1'. If they land near the left endpoint of A], then they will certainly separate on their next return, and so they will pick up no more distortion. It is not hard to see that the most distortion which can be introduced is given by f—’——( if? S z'K S K'a‘i = C(i)a"1. log EIT— Continuing this line of reasoning for any two points :1: and y in A,, we see that log f1 jzo z'KZa’j S K’a‘i = C(i)oz‘1. jEN f ’ f 3 -———-]_<_] We should point out that because of assumption (iv) above, these systems do not fit into the original theory in [You98]. This is because Young’s original theorem f’($ ) f’ (y) < T6i for some requires that there 18 some 6 < 1 such that, for all :1: ,y E A,, T > 0. Finally, we note that Condition (D) (See p. 18.) is clearly true, and Condition (E) is true by our choice of b since eC(‘)Q(z') S ea_ibo‘—i = (eb)a_i, and this decays faster than exponentially since eb < 1 and a < 1. Note that we have only verified the conditions for Theorem 2.3. With the amount of distortion present on each A,, we cannot verify the assumptions of Theorem 2.2. In particular, the sum ZEN 60(001 does not converge, but this is not important since 55 the existence of SRB measures for systems satisfying assumptions (ii) and (vi) above has been proved by Broise in [Br096]. 3.2 Finitely many domains of invertibility It is clear from the construction in the previous section that we may easily consider a function f : [0, 1] —> [0, 1] with finitely many domains of invertibility. Suppose now that we have some finite collection of a,s such that O < a1 < a2 < < a, < 1. We suppose that ID (i) f|[0,a1)($) 2 (TI, and we suppose that each subinterval (aj, aj+1) is partitioned (mod 0) into countably many subintervals A}, such that (ii) for each j and for each 1' E N, f IA- _ satisfies all the assumptions for f l A- in the .711 1 previous section mutatis mutandz’s with a replaced here by al. In particular, for every j we assume that fAJ-J- maps onto (a],a']"1). Also, we replace the assumption m (U A,) < b‘rl for some b < e‘1 by the corresponding assumption i>l m (UR,>1Ai) = m(Ui>1Aj.i) < b“—'. We define A = (a1,1). In each subinterval (a,, a,-+1), there is precisely 011 Aid which returns at time R,- = i, and since there are finitely many such subintervals, it is clear that the sums we consider in the previous example are all still finite here. 56 3.3 Countably many domains of invertibility In this section, we show how we can extend our work in the previous section to include maps with countably many domains of invertibility. As one imagines, the extension is not trivial. In fact, we require further knowledge of the rate of decay of the domains of invertibility. We assume that we have some countable collection of ais such that 0 < a1 < < a, < a,“ < 1. All of the assumptions in the previous section, including the assumption that m (U A“) < ba_l for some b < e", are taken here. Even in this i>l case, [Br096] proves the existence of SRB measures under assumptions (ii) and (vi), which we established at the beginning of this chapter; therefore, we only have to show that the sum ZEN eC(i)g(2') converges, where 9(2) 2 ba_i. First, note that (3.1) Z e50(i)g(2) = Z Z 650(i)g(2). iEN nEN 13531; Let us use Ej to denote the subinterval (aj,aj+1). In our construction, there is precisely one A, in each subinterval Ej which returns at time n, and C (2) is the same for all of these As with R,- = n. We see that we require sufficiently fast decay of the lengths of the intervals Ej in order to guarantee convergence of the sum (3.1). In particular, exponential decay will suffice. We assume that there is p < 1 such that IE3] S p7 ]E1|, where |Ej| denotes the length of the interval Ej. There is some constant K2 > 0 such that [AM] S Kgpl|ALn| for all j E N and for all n E N. Note that C(2) is constant on U iem A,, so without ambiguity we shall write C (n) for this Ri=n 57 constant. Then (3.1) is bounded above term-by—term by Z ESQ") IA1,n|K2 2 pi, nEN jEN and this is clearly finite since this is the same sum as we have in the system with only two domains of invertibility, except with a constant. multiple. 58 CHAPTER FOUR Newhouse-Jakobson maps We finally address the maps which motivate all our work. In [JNOO], Jakobson and Newhouse prove that certain types of piecewise smooth hyperbolic maps on the unit square in R2 have finite SRB measures. Furthermore, they prove that the natural extensions of the systems they consider are K—automorphisms; therefore, they are mixing, but the issue of the speed of mixing is left open. We make some meager progress in addressing this issue through the use of our theorem, but unfortunately we are not able to include the family of Newhouse-Jakobson maps in their full generality under the umbrella of our theorem. For the convenience of the reader, we give here a brief summary of some Newhouse—Jakobson maps which do fit into our theory, and we refer the reader to [J N00] for a discussion of them in their more general context. We assume that the unit square I2 is partitioned (modulo a set of measure zero) into countably many full-height curvilinear rectangles {Ei : 2 E N}. For convience we shall use the notation and vocabulary of [JNOO] and call E,- the 2th post. The upper 59 and lower boundaries of each E,- are subintervals of the upper and lower boundaries of I 2; the left and right boundaries of each E,- are the graphs of smooth functions 1132 dy 2,. We assume that F I E- extends to a 02 map on some neighborhood 8,- of E, and 1 1,-(y) such that l S a for some a 6 (0,1) which is independent of the function that S,- := F E,- C I 2 is a full-width strip. The left and right boundaries of each S,- are subintervals of the left and right boundaries of I 2; the upper and lower boundaries of dyi S a. For eachi E N, d2: each S, are the graphs of smooth functions y,(:1:) such that we do not permit the upper and lower boundaries of S,- to meet, nor do we permit the left and right boundaries of E,- to meet. For each 2 E N, we let f,- := F I E2 denote the restriction of F to the 2th post. There are some technical requirements on the 8,3 which are essential in the proof of the existence of SRB measures; these requirements are discussed in [JNOO], but they are not important to our work here since they do not enter into the discussion about the correlation functions. For each 2 E 12, let lz denote the horizontal line containing 2. We define the following: 62 (E,) = diam (12 (1 E1) 62,1nax : 1:133 62 (E2) (Si,min = min 52 (E2) 3 zEQ and we assume the following conditions on the geometry of the ES: (H1) int E,- flint Ej = (b if 2' # j, (H2) m (12 \ Um int E1) = 0, 60 (H3) — ZiEN 62.max 10g 62,min < 00. We assume also that Q := I2 \ UieN int E,- is hyperbolic for F; i.e., for each :1: E a? there is a splitting TIM = E; E) E: which varies coninuously with :1: E Q, a constant K0 > 1, and a Riemannian norm I - I such that 1. 0.192;) = E2...) and D.) 0 such that sup I f (2)] EN ifilx(z )| 26 _6z( Ei) 0, there is some compact set A C '7 and some constant K; = K4(A) > 0 such that 61 m7 (7 \ A) < e and such that, for all z E A and all n 2 0, (4.1) < K4, " deth“( (sz) I1&3th" (fJ(6 2)) where 6 is the stable holonomy map we introduced in Chapter 2. This is very nearly Condition (C) for C (2) = const independent of 2, but of course we have no idea how the ratio on the left-hand side of (4.1) behaves off the set A. Thinking of K 4 as a function of e, we then have to make some comparatively strong assumption on the regularity of the stable foliation [‘5 so that we can guarantee how K4 will behave as m, (7 \ A) —> 0. One way to do this is to assume that det Df“ is Lipschitz; i.e., (4-2) Idet Dfu(1‘0) — det Dfu(fl/0)l S Kslxo — gel for some K 5 > 0 whenever yo 6 73(20). Note that det Df“(:1:) _ 1+ det Df“($) —— det Df“(y) < ex (det Df“($) —' det Df“(y)) det D f“(y) ‘ det D f“(y) — p deth“(y) ° Thus, if y E 73(23), we have log 10—01 = i log j=o i=0 1 u j u ' Sgldewfumyn 'lde‘Df (f xl‘deth (ny)| oo 2: |det D f“( fJx) — det Df"(f”y)| i=0 K . . S RE 2 lf’w - f’yl det Df“(fj:r) (f. ) det Df“(fj:1:) det Df“ ( dethu ny) y__; <__ _Ko 8 1:0 K °° 1 J Six—ylZ(R—) 0 j=0 0 62 =2 K6, since :1: and y are in the same stable leaf. Similarly, for each n E N we have deth"( °° deth“( f J?) lOgH deth“(f =210gdeth“ fy) 5 0° 1 J S :)I_(—lI—y|JZ-;(T{—) Thus, the system satisfies Condition (C) for C(2) 2 K6 for all 2 E N and for o: 2 K51. As we (lid in the previous two chapters, let us take A = U22 E,. As with the previous examples, this will allow us to define the A,-s so that each E,- has precisely one A,- which returns at time n for each n E N. Furthermore, because F maps each full-height. post Ej onto a full—width strip 33-, it will be very easy for us to define the As explicitly. In more general hyperbolic systems in which full-height sets might be mapped across some proper subset of the posts, constructing the As is significantly more difficult. Let us fix a post E,. We shall define Am to be that part of E,- which is mapped across A by F; i.e., A“ = fflA. We define A132 to be that part of E,- which is first mapped to E1 by F and then across A; i.e., Am 2 ff] ff 1A. We continue inductively so that we may define Aid- 2 ff] f1— j +1A. We do this for each 2 E N. Then AM is that part of E, which returns to A at time j. The final condition we must verify is that there is C > 0 and 00 < 1 such that m7 {1‘ 6 70A : R(.I:) >1} 3 066 for all ”y E F“ and for all I E N. For general 63 hyperbolic systems, this requires a bit of work to prove; however, for the system we are considering here, this is easily seen to be true since each full—height post E, is mapped onto a full-width strip 8,. This is clear from our construction of the Aid-s above. Let ’7 E PuflA be a full-width u—disk in A, and let '7' E ImflAid- be its preimage in A133”; i.e., F 377’ = '7. Then by the hyperbolicity of F, the length of 7’ is no greater than KO—j - 62.max- As we did in Section 3.3, since we have countably many domains of i1’1vertibility, we assume that there is some p < 1 such that 62,max < p‘. Then my {1: E '7 H A : R(:1:) 2 j} S ng 21ers 5mm, which is what we wanted to verify. Notice that. we did not worry about verifying the conditions of Theorem 2.2 to determine existence of SRB measures; however, this is not important here since New- house and J akobson prove the existence of SRB measures whose conditional measures on unstable leaves are equivalent to Lebesgue measure, and this is precisely what we require in order to apply Theorem 2.3. Furthermore, they prove that the natural extension is a K-automorphism; therefore, the dynamical system we consider here is exact and, thus, mixing. 64 Auto-correlation function, 4 Axiom A, 2 Correlation function, 4 Entropy Metric, 2 Hyperbolic product structure, 14 Independent random variables, 4 Induced map, 10 Invariant measure, 1 J acobian Unstable, 17 K-automorphism, 59 Lasota—Yorke inequality, 33 Lyapunov exponent, 2 Measure Invariant, 1 Physical, 1 SRB, 2 Mixing, 3, 6 Topological, 4 Observable, 1 Pesin’s Formula, 2 Return map, 10 Return time, 9, ll Separation time, 19 SRB measure, 2 Topological mixing, 4 Tower, 10, 15 Tower map, 10 INDEX [ALP] [BalOl] [BMD] [Br096] [C he99] [D858] [Hu] [HY95] [muse] [JNom [Kac47] BIBLIOGRAPHY J. F. Alves, S. Luzzatto, and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, arXiv preprint arXiv:math.DS/0205191. V. Baladi, Decay of correlations, Proc. Sympos. 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