The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-090 Wien, Austria Statistical Properties of Dynamical Systems With Some Hyperbolicity Lai{Sang Young Vienna, Preprint ESI 445 (997) May 7, 997 Supported by Federal Ministry of Science and Research, Austria Available via

2 STATISTICAL PROPERTIES OF DYNAMICAL SYSTEMS WITH SOME HYPERBOLICITY Lai-Sang Young* Department of Mathematics University of California, Los Angeles Los Angeles, CA April 996 This paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts (though not necessarily all) of their phase spaces. The main results are for discrete time systems. To put this work into context, recall that for Axiom A attractors the picture has been fairly complete since the 970's (see [S], [B], [R2]). Since then the development has, by and large, been two-pronged: there is a general nonuniform theory that deals with properties common to all dieomorphisms with nonzero Lyapunov exponents ([O], [P], [Ka], [LY]), and there are detailed analyses of specic kinds of dynamical systems including, for example, billiards, -dimensional and Henon-type maps ([S2], [BSC]; [HK], [J]; [BC2], [BY]). Statistical properties such as exponential decay of correlations are not enjoyed by all dieomorphisms with nonzero Lyapunov exponents. In this paper I will attempt to understand these and other properties for a class of dynamical systems larger than Axiom A. This class will not be dened explicitly, but it includes some of the much studied examples. By looking at regular returns to sets with good hyperbolic properties, one could give systems in this class a simple dynamical representation. Conditions for the existence of natural invariant measures, exponential mixing and central limit theorems are given in terms of the return times. These conditions can be checked in concrete situations, giving a unied way of proving a number of results, some old and some new. Among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of Henon-type maps. The dynamical picture we wish to focus on is the following. Let f be the map in question, and suppose that f admits a \horseshoe" with innitely many branches and variable return times. More precisely, has a product structure in the sense *This research is partially supported by NSF Typeset by AMS-TEX

3 2 that it is the intersection of two transversal families of stable and unstable manifolds. Dynamically, it is the disjoint union of a countable number of sets i with the property that each i extends fully in the stable direction, and for each i there is an integer R i such that f R i maps i onto a subset of, crossing it completely in the unstable direction. Let R be the return time function, i.e. R j i = R i. We prove the following: () if intersects its unstable manifolds in positive Lebesgue measure sets, and R Rd u < where u denotes Lebesgue measure on unstable manifolds, then f admits a Sinai-Ruelle-Bowen measure ; and (2) if additionally u fr > ng decreases exponentially with n, then (f; ) has exponential decay of correlations for Holder continuous test functions provided that the usual aperiodicity conditions are met; under the same conditions the central limit theorem also holds. Conceptually, (2) says that under the usual aperiodicity assumptions, a sucient condition for exponential mixing is that \most" arbitrarily small pieces of unstable manifolds grow to a xed size at exponential speeds. Precise formulations of these results are given in Section. We remark that our setup bears a certain resemblance to countable state Markov chains for which the corresponding results are also valid. We must emphasize, however, that these results are for discrete time systems; (2) above is false for ows. In order to apply these \abstract" results to specic dynamical systems, we must ask the following questions: given f with some hyperbolicity, does with u () > 0 exist, how to nd it, and how to determine the nature of R? We do not know how to deal with general dieomorphisms, so let us specialize to the following situation: suppose there is a recognizable set away from which f is uniformly hyperbolic, and suppose that when an orbit passes near this set it suers a certain setback in its hyperbolicity from which it will attempt to recover. Assume further that we have quantitative knowledge of both the setback and recovery. The methods of this paper will suggest that under these conditions (a) there is a systematic way of choosing, namely by xing a box, taking points in it that approach the \bad set" not faster than a certain rate, and running the system until the various parts of return as desired; (b) the speeds with which orbits recover from the inuence of the \bad set" are reected in u () and in the nature of the return time function R. To give some examples of \bad sets", for billiards they might be thought of as directions that give rise to trajectories making tangential contacts with the boundary of the table, whereas for Henon-type maps it is clearly the \turns" that spoil hyperbolicity. Our scheme of proof is potentially applicable to dynamical systems for which the mechanisms that cause hyperbolicity to fail are known and the source of nonhyperbolicity is localized. In Part I of this paper we will prove () and (2) assuming the existence of a \horseshoe" with innitely many branches and variable return times. In Part II we will illustrate (a) and (b) for several relatively simple situations. In each case it will

4 3 be shown that the return time function has the desired exponential estimate. It then follows immediately from the results of Part I that they admit SRB measures, have exponential decay of correlations etc. TABLE OF CONTENTS Part I. An abstract model and its mixing properties. Setting and assertions 2. Sinai-Ruelle-Bowen measures 3. Spectral gap of an associated operator 4. Exponential decay of correlations 5. Central limit theorem Part II. Examples of systems that t this model 6. Axiom A attractors 7. Piecewise hyperbolic maps 8. Billiards with convex scatterers 9. Logistic maps 0. Henon-type attractors The work discussed in Section 0 is joint with M. Benedicks; the proofs will appear elsewhere. Acknowledgements. Some of the ideas of this paper were conceived while I was working on the Henon maps and I would like to thank Benedicks for his input. My thanks go also to N. Chernov for sharing with me his expertise on billiards as well as his ideas on decay of correlations. Finally, I wish to acknowledge the hospitality of MSRI, where much of this work was done. PART I. AN ABSTRACT MODEL AND ITS MIXING PROPERTIES. Setting and Assertions Let f : M be a C +" dieomorphism of a nite dimensional Riemannian manifold M. In applications we will allow f to have discontinuities or singularities, but these \bad" parts will not appear in the picture we are about to describe. Thus as far as Part I is concerned we may assume that f and f? are dened on all of M. Let d(; ) denote the distance between points. Riemannian measure on M will be denoted by ; and if W M is a submanifold, then W denotes the measure on W induced by the restriction of the Riemannian structure to W. The basic object

5 4 of interest here consists of a set M with a \hyperbolic produce structure" and a return map f R from to itself. Precise denitions are given in. and.2; the required properties are listed in (P)-(P5); and the main results of Part I are stated in.4... A \horseshoe" with innitely many branches and variable return times. We begin with some formal denitions. An embedded disk M is called an unstable manifold or unstable disk if 8x; y 2 ; d(f?n x; f?n y)! 0 exponentially fast as n! ; it is called a stable manifold or stable disk if 8x; y 2 ; d(f n x; f n y)! 0 exponentially fast as n!. We say that? u = f u g is a continuous family of C unstable disks if the following hold: * K s is an arbitrary compact set; D u is the unit disk of some R n ; * u : K s D u! M is a map with the property that - u maps K s D u homeomorphically onto its image, - x 7! u j (fxg D u ) is a continuous map from K s into Emb (D u ; M), the space of C embeddings of D u into M, - u, the image of each fxg D u, is an unstable disk. Continuous families of C stable disks are dened similarly. Denition. We say that M has a hyperbolic product structure if there exist a continuous family of unstable disks? u = f u g and a continuous family of stable disks? s = f s g such that (i) dim u + dim s = dim M; (ii) the u -disks are transversal to the s -disks with the angles between them bounded away from 0; (iii) each u -disk meets each s -disk in exactly one point; and (iv) = ([ u ) \ ([ s ). We will assume throughout Part I that (P) there exists M with a hyperbolic product structure and with f\g > 0 for every 2? u. Next we dene a return map on that gives it the structure of a \horseshoe" { except that unlike the standard horseshoe this one has innitely many branches returning at variable times. Let? u and? s be dening families for. A subset 0 is called an s-subset if 0 also has a hyperbolic structure and its dening families can be chosen to be? u and? s 0 with? s 0? s ; u-subsets are dened analogously. For x 2, let u (x) denote the element of? u containing x. We assume: (P2) There is a countable number of s-subsets ; 2 ; : : : s.t. - on each u -disk, uf(? [ i ) \ u g = 0; - for each i; 9R i 2 + s.t. f R i i is a u-subset of ; we require in fact that for all x 2 i ; f R i ( s (x)) s (f R i x) and f R i ( u (x)) u (f R i x); - for each n, there are at most nitely many i's with R i = n;

6 5 - min R i some R 0 depending only on f. (R 0 depends in fact only on the constants C and in (P3)-(P5).).2. Separation times and derivative estimates. For every pair x; y 2, we assume there is a notion of separation time denoted by s 0 (x; y). If s 0 (x; y) = n, then the orbits of x and y are thought of as being \indistinguishable" or \together" through their n th iterates, while f n+ x and f n+ y are thought of as having been \separated." The nature of the separation will not be specied in this abstract part. In applications, this could be when the two points have moved a certain distance apart, or have landed on opposite sides of a discontinuity curve, or that their derivatives have ceased to be comparable. Here we assume only that (i) s 0 (; ) 0 and depends only on the s -disks containing the two points; (ii) the number of \distinguishable" n-orbits starting from is nite for each n; (iii) for x; y 2 i ; s 0 (x; y) R i + s 0 (f R i x; f R i y); We remark that in the proofs to follow, it is only necessary that (ii) holds for n R 0. (See the remark at the end of 3.5.) We now state the required analytic estimates that accompany the topological picture in.. Let f u denote the restriction of f to u -disks, and let det(df u ) be the Jacobian of D(f u ). We assume there exist C > 0 and < s.t. the following hold for all x; y 2 : (P3) Contraction along s -disks. For y 2 s (x); d(f n x; f n y) C n 8n 0. (P4) Backward contraction and distortion along u. For y 2 u (x) and 0 k n < s 0 (x; y), we have (a) d(f n x; f n y) C s 0(x;y)?n ; (b) log ny i=k det Df u (f i x) det Df u (f i y) Cs 0(x;y)?n : (P5) Convergence of D(f i j u ) and absolute continuity of? s. (a) For y 2 s (x), log Y i=n det Df u (f i x) det Df u (f i y) Cn 8n 0: (b) For ; 0 2? u, if : \! 0 \ is dened by (x) = s (x) \ 0, then is absolutely continuous and d(? 0) d (x) = Y i=0 det Df u (f i x) det Df u (f i x) :

7 6 (In practice, (P5)(b) is usually a consequence of (P3)-(P5)(a). To make this a logical implication in an abstract setting, however, requires more technical formulation of the other conditions than we would like to give.) (P3)-(P5) are standard for Axiom A attractors. We wish to stress however that they are strictly less stringent than uniform hyperbolicity: we allow oscillatory behavior along s as long as the cumulative contraction starting from is uniform, and the backward contraction conditions along u are imposed only at certain checkpoints allowing for a variety of behaviors in between. This is what allows us to include, for example, the Henon maps..3. A Markov extension of f. Our next step is to construct an extension of f : [ n0 f n which has on it a natural Markov partition with a countable number of states. By an extension of f : [f n we refer to a dynamical system F : for which there is a projection map :! [f n satisfying f = F. In general will not be?. Let R :! + be the return time function, i.e. Rj i = R i, and let f R : denote the return map with f R j i = f R i ji. In ergodic theory there is a standard construction called a special ow built over a map under a function. Our extension F : will be the discrete time version of the special ow built over f R : under R. More precisely, let and dene def = f(x; `) : x 2 ; ` = 0; ; : : : ; R(x)? g (x; ` + ) if ` + < R(x) F (x; `) = (f R x; 0) if ` + = R(x): An equivalent but less formal way of looking at is to view it as the disjoint union [ `=0 ` where ` consists of those pairs (x; `) 2 the second coordinate of which is `. We will refer to ` as the `th level of the tower. Clearly, ` is a copy of fx 2 : R(x) > `g; we will sometimes confuse it with this set and overlook the hopefully benign abuse of language. It is also clear that ` can be identied with the union of a collection of i 's. We wish to dene a Markov partition D = f`;j g for F :. If R is unbounded as it usually is, this partition will necessarily have an innite number of states. Let us agree to take each `;j to be an s-subset of `. It will be important for our purposes to choose D in such a way that points in each `;j are still \together", meaning that if (x; `); (y; `) 2 `;j, then s 0 (x; y) `. >From what we have said so far, a natural candidate for D is the partition of each ` into i 's. We will see in Section 3, however, that this partition is \too big", and that it is convenient to have no more than a nite number of `;j 's on each level of the tower. Let D` denote Dj`. We will regard ` as a subset of in this paragraph and choose the D`'s to have the following properties: (a) D` has only a nite number of elements each one of which is the union of a

8 7 collection of i 's; (b) each element of D` is contained in an element of D`? ; (c) if x and y belong in the same element of D`, then s 0 (x; y) `; (d) if R i = R j for some i 6= j, then i and j belong in dierent elements of D Ri?. The construction of fd`g is in fact quite arbitrary. Let D 0 consist of a single element. We will go from one stage to the next rening the D`'s subject to the conditions above. That (c) is compatible with never breaking up the i 's follows from the fact that s 0 (x; y) R i for x; y 2 i (property (iii) of s 0 (; )). As for the niteness of each D`, it is compatible with (c) and (d) by property (ii) of s 0 (; ) and the fact that there are at most nitely many i's with the same R i. To repeat then, we have constructed a Markov partition D = f`;j g for F : the elements of which are copies of s-subsets of. The image of each `;j under F is a nite union of `+;j 0's together with possibly one u-subset of 0. Let `;j = `;j \ F? ( 0 ). We think of `;j? `;j as \moving upward" under F, while `;j returns to the base. Note that when `;j 6=, it is in fact a copy of one of the i 's. Observe also that F is? on? `;j, but that the images of the [`;j `;j 's could overlap. Given this construction it is now natural to redene the separation time of x; y 2 u \ `;j to be s(x; y) def = the largest n s.t. for all i n; F i x and F i y lie in the same element of f`;j g: We claim that (P4) is valid for x; y 2 u \ `;j with s(; ) in the place of s 0 (; ). To verify this, rst consider x; y 2. We claim that s(x; y) s 0 (x; y). If x; y do not belong in the same i, then this follows from rule (c) in the construction of D`; if x; y 2 i but f R x; f R y are not contained in the same j, then s(x; y) = R i + s(f R x; f R y), which is s 0 (x; y) by property (iii) of s 0, and so on. In general, for x; y 2 `;j, let x 0 = F?`x; y 0 = F?`y be the unique inverse images of x and y in 0. Then by denition s(x; y) = s(x 0 ; y 0 )? `, and what is said earlier on about x 0 and y 0 is equally valid for x and y. From here on s 0 (; ) is replaced by s(; ) and (P4) is modied accordingly. The two views of F : that we have presented can be summarized as follows. One is to regard it as a special ow over the \horseshoe" map f R : under the return time function R. The other is to view it as the combinatorial object given by the directed graph whose vertices correspond to f`;j g. In this graph each vertex moves upward, branching where separation occurs { except that at many vertices there is also the possibility of returning to 0, the \root" of the tree..4. Statements of theorems. First we give some relevant facts and denitions.

9 8 Denition 2. An f-invariant Borel probability measure on M is called a Sinai- Ruelle-Bowen (SRB) measure for f if f has a positive Lyapunov exponent? a.e. and the conditional measures of on unstable manifolds are absolutely continuous with respect to the Riemannian measures on these manifolds. Let us restrict ourselves to systems with no zero Lyapunov exponents. If f is conservative, i.e., if it preserves a measure equivalent to the Riemannian measure on M, then from the point of view of physical observations is the most natural invariant measure { and it is a special case of an SRB measure. For dissipative systems, SRB measures are, in some sense, the only invariant measures that are physically observable: if is SRB, then there is a positive -measure set consisting of points that are -generic, i.e. n n? i=0 'f i (x)! R 'd for all continuous ' : M! R. (See [PS].) If is equivalent to Riemannian volume or is SRB, and there are no zero Lyapunov exponents, then the phase space is decomposed into mixing components as follows: (f; ) has at most a countable number of ergodic components supported on, say, X ; X 2 ; X 3 ; ::: ; for each i, either f : (X i ; jx i ) is mixing or X i is further decomposed into a nite cycle, i.e. X i = Xi [ [ X N i i with fxi j = Xj+ i and fx N i i = Xi, and f N i : (Xi j ; jxj i ) is mixing. Next we turn to the speed of mixing. Denition 3. Let be an f-invariant Borel probability measure and let F be a class of functions on M. We say that (f; ) has exponential decay of correlations for functions in F if 9 < s.t. 8'; 2 F; 9C = C('; ) s.t. (' f n ) d? 'd d C n 8n : Let (f; ) be as above, and let ' : M! R. Consider the random variables '; ' f; ' f 2 ; : : : on the probability space (M; ). Then the exponential decay of correlations for (f; ) says in particular that ' f n and ' become uncorrelated exponentially fast in n. One could ask about other limit theorems. Denition 4. Consider ' with R 'd = 0. We say that ' satises the Central Limit Theorem with respect to (f; ) if the above random variables do, i.e. if for some 0. X n? p ' f i n i=0 distr?! N (0; ) We now state the main results of Part I. All notations are as in.-.3, and (P)-(P5) are assumed.

10 9 Theorem. If for some 2? u \ Rd < ; then f admits an SRB measure. Recall from (P5) that for all 2? u ; j (\) are uniformly equivalent. Hence the integrability condition above is equivalent to that on all u -disks. Let H denote the class of Holder continuous functions on M with Holder exponent, i.e. H := f' : M! R j 9C > 0 s.t. j'(x)? '(y)j Cd(x; y) 8x; y 2 Mg; and let be the SRB measure given by Theorem. Theorem 2. Suppose that (a) 9C 0 > 0 and 0 < s.t. for some 2? u, fx 2 \ : R(x) > ng C 0 n 0 8n 0; (b) (f n ; ) is ergodic 8n. Then (f; ) has exponential decay of correlations for functions in H for every > 0, with = (). Theorem 3. Under the hypotheses of Theorem 2, every ' 2 H with R 'd = 0 satises the Central Limit Theorem wrt (f; ), with = 0 i ' = f? for some 2 L 2 (). We prove Theorems -3 by working with F :. In particular the corresponding results hold for F. For a precise description of the class of functions on to which Theorems 2 and 3 apply, see Sections 3 and 4. Theorems -3 also have analogs in the setting of countable state Markov chains. For example, a simplied version of our proofs gives the following result which in all likelihood is known (and in any case is not hard) but for which I have not been able to locate a reference: Theorem. Let X ; X 2 ; : : : be a stationary Markov chain on the state space S = f0; ; 2; : : : g. Assume the usual ergodicity and aperiodicity conditions. Suppose also that there exist C > 0 and < such that for all n >, P (X = 0; X i 6= 0 for i = 2; : : : ; n) < C n : Then there exists < such that for all bounded ' : S! R, je('(x )'(X n ))? (E'(X )) 2 j C(') n 8n :

11 0 2. Sinai-Ruelle-Bowen Measures Our proof of Theorem consists of the following 3 steps: () We construct an f R -invariant nite Borel measure 0 on with absolutely continuous conditional measures on u -leaves; clearly 0 can be identied with an F R -invariant measure ~ 0 on 0. (2) We extend ~ 0 to a nite F -invariant Borel measure ~ on. (3) We verify that := ~()? ~ has the SRB property. To construct 0, we x an arbitrary u -leaf, call it 0, and let 0 := 0 j ( 0 \). (P) says that 0 > 0. Let j be the densities of the conditional measures of (f R ) j 0 on u -leaves, i.e. j = d(f R ) j 0 d =((f R ) j 0 )() whenever ((f R ) j 0 )() > 0. It follows from (P4)(b) that j (x) j (y) expfcs(x;y) g 8x; y 2 \ : In particular, 9M 0 > 0 independent of j or s.t. (*) M? 0 j M 0 on \ while j 0 on?. Let 0 be an accumulation point of j=0 (f R ) j 0 n=;2;::: in the weak -topology, and let f 0 g be the conditional measures of 0 on u -leaves. We claim that 0 for a.e. with uniform bounds M? d 0 d M. To see this, x an arbitrary open set! with 0 (@!) = 0 and let S! denote the s-subset of corresponding to!. Also x a u-subset U that is a compact neighborhood of. Then () together with (P5) imply that for all j, (**) M? (S! ) () ((f R ) j 0 )(U \ S! ) ((f R ) j 0 )(U) n n? M (S! ) () for some M. (A stronger version of (P5)(b) would have allowed us to take M near M 0 for U suciently thin.) The bounds in () are passed on to 0. By taking U arbitrarily small, the martingale convergence theorem allows us to conclude that M? (S! ) () 0 (S!) (S! ) M () for a.e.. Since! is arbitrary, the density statement for 0 For (2), let ~ := X j=0 F j (~ 0 j fr > jg); follows.

12 and observe that since R 0 ~() is equivalent to Rd \ <. is uniformly equivalent to j ( \ ), the niteness of As for (3), the f-invariance of is evident. The SRB property is also obvious since f j 0 clearly has absolutely continuous conditional measures on ff j u g for every j, and these are unstable manifolds. 3. Spectral Gap of an Associated Operator The purpose of this section is to introduce a Perron-Frobenius operator or transfer operator associated with F : and to prove the existence of a spectral gap under the usual aperiodicity conditions. What exactly this tells us about stochastic processes generated by f : (M; ) will be explained in Sections 4 and Reduction of F : to an \expanding map". Let := = where x y i y 2 s (x). Since F takes s -leaves to s -leaves, the quotient dynamical system F : is clearly well dened { topologically at least. The purpose of this subsection is to study the dierential properties of F in the sense of the Jacobian of F with respect to a reference measure. Consider in general a measurable bijection T : (X ; m )! (X 2 ; m 2 ) between two nite measure spaces. We say that T is nonsingular if it maps sets of m -measure 0 to sets of m 2 -measure 0. If T is nonsingular, we dene the Jacobian of T wrt m and m 2,? d(t written J m ;m 2 (T ) or simply J(T ), to be the Radon-Nikodym derivative m 2) dm. To introduce a \dierential structure" for F : in the sense above, it suces to dene a reference measure m on := = in a way that J m; m (f R ) makes sense. (Notations such as f R : and = ` etc. are given the obvious meanings.) [` We then let m j ` be the measure induced from the natural identication of ` with a subset of 0, so that J( F ) except on F? ( 0 ), where J( F) = J(f R F?(R?) ). We will continue to use det(df u ) to denote J(f) wrt. We now dene m on following ideas that have been used for Axiom A (see e.g. [B]). Fix an arbitrary ^ 2? u. For x 2, let ^x denote the point in s (x) \ ^, and dene u n (x) = n? X i=0 ('(f i x)? '(f i^x)) where '() = log j det Df u ()j. From (P5)(a) it follows that u n converges uniformly to some function u. On each 2? u, we let m be the measure whose density wrt is e u I \ where I () is the indicator function. Clearly, f R i j (i \ ) is nonsingular wrt these reference measures. If f R i ( i \ ) 0, then for x 2 i \ we write J(f R )(x) = J m ;m 0 (f R i j (i \ ))(x). Lemma. () Let ; 0 : \! 0 \ be the sliding map along? s. Then m = m 0. (2) J(f R )(x) = J(f R )(y) 8y 2 s (x).

13 2 (3) 9C > 0 s.t. 8i and 8x; y 2 i \, J(f R )(x) J(f R )(y)? C 2 s(f R x;f Ry) : Lemma () allows us to dene m on to be the measure whose representative on each 2? u is m. Statement (2) says that J(f R ) is well dened wrt m, and (3) says that log J(f R ) has a dynamically dened Holder type property, in the sense that s(f R x;f R y) could be viewed as a notion of distance between f R x and f R y (see (P4)). Proof of Lemma. () Suppose x = x 0. Then the density of m wrt 0 at x 0 is e u(x) d( ) d, and the second factor is = e u(x0 )?u(x) by (P5)(b). 0 (2) For? a.e. x 2 \, we have J(f R )(x) = j det D(f R ) u xj e u(f R x) e?u(x) : We verify that J(f R )(x) depends only on ^x and not on x itself: log J(f R )(x) = = R? X i=0 R? X i=0 '(f i x) +? '(f i ^x) + X X i=0 i=0 X i=0 '(f i (f R x))? '(f i ( f d R x)) ('(f i x)? '(f i ^x)) '(f i (f R ^x))? '(f i ( f d R x)) : (3) We estimate ju(x)? u(y)j as follows. Let k s(x; y). Then 2 ju(x)? u(y)j + k? X i=0 X i=k ('(f i x)? '(f i y))? ('(f i x)? '(f i^x))? k? X i=0 X i=k ('(f i^x)? '(f i ^y)) ('(f i y)? '(f i ^y)) : Using (P4)(b) for the rst two sums and (P5)(a) for the latter two, we obtain ju(x)? u(y)j 2C s(x;y)?k + 2C k 4C 2 s(x;y). Now log J(f R )(x) J(f R )(y) = log det D(f R ) u (x) det D(f R ) u (y) + (u(f R x)? u(f R y))? (u(x)? u(y)): The rst term is C s(f R x;f R y) by (P4)(b). The second and third have been estimated above.

14 3 The Perron-Frobenius operator will shall introduce will in fact be associated with F : and will act on a suitable Banach space of functions dened on. If the spectral properties of this operator are as desired, then it will have an eigenfunction corresponding to the eigenvalue ; in other words, our function space must contain an element such that d m is an invariant measure for F. In part to motivate the choice of this function space, we rst discuss the regularity of. Now we have already encountered one invariant density in Section 2 obtained by collapsing ~ along stable disks. In the next lemma we give an alternate construction. Lemma 2. R Assume Rd m < and let 2. Then F : has an invariant probability measure of the form d = d m where satises c 0 c? 0 for some c 0 > 0 and j(x)? (y)j C s(x;y) 8x; y 2 `;j : Proof. We construct by realizing it as the density wrt m of an accumulation point n? F ( i mj 0 ). Let us consider rst n j 0. Let n be the density of n i=0 of n := n wrt m. Then n j 0 = n j j n where j n is the density of F i ( mj j ) and the j 's range over all components of F?i ( 0 ) \ 0 ; i n. The variation of each j n is estimated as follows: Let x; y 2 0, and let x 0 ; y 0 2 j be s.t. F i x 0 = x; F i y 0 = y. Then j n(y) j n(x) = J F i (x 0 ) J F i (y 0 ) = qy k= J F( F i k? x 0 ) J F( F i k? y 0 ) where i < i 2 < < i q = i are the times when F p j 0, and by Lemma. Thus J F( F i k? x 0 ) J F ( F i k? y 0 ) expfc s( F ik x 0 ; F ik y 0 ) g expfc (i?i k)+s(x;y) g j n(y) j n(x) expfc 0 s(x;y) g; an estimate that is easily seen to be valid also for n. To nish we must let n!. Partitioning 0 successively into sets with the property that x; y in distinct sets satisfy s(x; y) k; k = ; 2; : : : ; we obtain the corresponding distortion estimate for. Reasoning as in the proof of Theorem, we see that f n g has an accumulation point on with 0 < ( ) <. Thus we have that on 0 ; c 0 c? 0 and For x; y 2 `;j, use j ` = F?`. j(x)? (y)j jj (x) (y)? C s(x;y) : We assume for the rest of Part I that fr ng C 0 n 0 for every 2?u.

15 Choice of function space and denition of Perron-Frobenius operator. Let F : and m be as in the last subsection, and let f `;j g be the Markov partition for F : corresponding to f`;j g. We collect below some important facts about ( F ; ; m) and f`;j g. All have been introduced or proved before except for (I)(i), which is formulated precisely for the rst time here. Let be s.t. 2 <, and let C be as in Lemma (3). (I) Height of tower. (i) R N for some N satisfying C e C N 00 ; (ii) mfr ng C 0 0 n 0 8n 0 for some C 0 0 > 0 and 0 <. (II) Regularity of the Jacobian. (i) J F on? F? ( 0 ), (ii) J F (x) J F (y)? C s( F x; F y) 8x; y 2 `;j : We explain the reason for (I)(i). The exponent of is decreased by with each step up the tower; we want this gain to outdo the constants due to nonlinearities, and (I)(i) guarantees this between consecutive returns to 0. We now choose a function space suitable for our purposes. Let " > 0 be s.t. (i) e 2" 0 <, (ii) m( 0 )? `;j m( `;j )e`" 2. Note that (ii) is consistent with `;j m( `;j ) = m( 0 ), and property (I)(ii). remark also that should be thought of as < e?", because N 00 We while (ii) above implies that e?"n 2. Let X = f ' :! C j k'k < g where k k is dened as follows. We write '`;j = ' j `;j, and let jj p denote the L p -norm wrt the reference measure m. Then k 'k := k 'k + k 'k h where k 'k := supk '`;j k ; `;j and k '`;j k and k '`;j k h are dened by k 'k h := supk '`;j k h ; `;j k '`;j k := j '`;j j e?`" ;! j '(x)? '(y)j k '`;j k h := ess sup e?`" : x;y2 `;j s(x;y) It is straightforward to verify that (X; k k) is a Banach space. Note that, the invariant density of F, is an element of (X; k k).

16 5 We record for future use the following relation: 9C 00 0 s.t. 8 ' 2 X, j 'j C 00 0 k 'k : This is true because and so j '`;j j j '`;j j m( `;j ) k '`;j k e`" m( `;j ); X X j 'j k 'k m( `)e`" k 'k ` ` C 0 0`0e "` < : The Perron-Frobenius operator or transfer operator associated with the dynamical system F : and reference measure m is dened to be P ( ')(x) = X y: F y=x '(y) J F(y) : The next few subsections are about the spectral properties of P as an operator on the function space (X; k k). To distinguish between F : and its quotient system F :, we have, up until now, used bars () to denote points, subsets and functions of the latter. The rest of Section 3 will be exclusively about F :, and for the sake of notational simplicity, we will drop all the bars Outline of proof of spectral gap. Our main result is Proposition A. () P is a bounded linear operator on (X; kk); its spectrum (P ) is contained in fjj g; and 9 0 < s.t. (P ) \ fjj 0 g consists of a nite number of points the eigenspaces corresponding to which are all nite dimensional. (2) If the greatest common divisor (gcd) of fr(z) : z 2 0 g is =, then is the only point of (P ) on fjj = g and it is a simple eigenvalue, i.e., its eigenspace is -dimensional. Our proof of () follows a standard route. The two main ingredients are (i) contractivity and (ii) approximation by an operator of nite rank. These two properties are made precise in Lemmas 3 and 4 below; their proofs are given in 3.4 and 3.5. Lemma 3. (a) P (X) X, and P : X! X is a bounded operator. (b) 9K > 0 s.t. 8' 2 X, where N and " are as in 3.2. kp N 'k e?"n k'k + Kj'j

17 6 Corollary to Lemma 3. (P ) fjj g. Lemma 4. Let 0 be s.t. e?"n < N 0 Q : X! X s.t. <. Then there exists a nite rank operator kp N? Qk < N 0 : Lemma 4 implies the quasi-compactness of P. (See e.g. [D&S] VIII.8.) The main property of F : used in the proof of Proposition A(2) is the following. Let be the F -invariant measure whose density is (see e.g. Lemma 2). Then: Lemma 5. If gcdfr(z) : z 2 0 g =, then (F; ) is exact, which in this case is equivalent to (F n ; ) being ergodic for all n. Aside from the fact that needs to be bounded away from 0, the conclusion of (2) follows from the exactness of (F; ) and general principles not specic to the present setting. This will be explained in Contractivity of P. The aim of this subsection is to prove Lemma 3 and its corollary. We distinguish between the cases ` N and ` < n. For x 2 ` with ` N; F?N fxg consists of a single point fyg and JF N (y) = ; hence the estimates are quite trivial. On `;j with ` < N; F?N has innitely many branches; they originate from distinct `;j 's, and each passes through 0 exactly once. Estimate #. For ` N, k(p N ')`;j k e?"n k'k : Proof. k(p N ')`;j k def = j(p N ')`;j j e?`"! = ess sup j'yje?(`?n)" e?"n y2f?n `;j k'k e?"n : Estimate #2. 9K > 0 s.t. 8` with 0 ` < N and 8', Proof. We x `; j and estimate k k by (*) k(p N ')`;j k X br k(p N ')`;j k K j'j + 2e C N k'k h : JF N I F?N `;j 'IF?N `;j

18 7 where \ br " means summing over all inverse branches of F?N and I () is the indicator function. We further split this sum into two sums, () and (2), corresponding to estimating each branch of j'i F?N j `;j by 'IF?N `;j m(f?n `;j ) Using the distortion estimate we obtain that JF N I F?N `;j () X br e C F?N `;j 'dm + e C m(f?n `;j ) m(`;j ) 'IF?N `;j ess sup j'y? 'y 2 j: y ;y 2 2F?N `;j m(`;j) ; K j'j where K := e C maxfm(`0;j 0)? : `0 < N; all j 0 g is nite because there are only nitely many `0;j 0's with `0 < N. To estimate (2), let `br be the level of the branch of F?N `;j in question. Let br = `0;j, where 0 `0;j0 is the component on the top level of through which this branch passes, and let `br = `0. Then (**) (2) " X br e C k'k h N JF N I F?N `;j ess sup y ;y 2 2F?N `;j! X m e`br br " m 0 br j'y? 'y 2 j s(y e?`br" ;y 2 )! e`br" In the rst inequality above we have used the fact that 8y ; y 2 2 F?N `;j ; s(y ; y 2 ) N, and in the second we have used the distortion estimate JF N I F?N `;j e C m( br ) The quantity in parenthesis in (**) is clearly 2. m( 0 ) : # N Estimate #3. For ` N, k(p N ')`;j k h N e?"n k'k h : Proof. k(p N ')`;j k h def = ess sup x ;x 2 2`;j! j(p N ')x? (P N ')x 2 j s(x ;x 2 e?`" )! j'y? 'y 2 j = ess sup y ;y 2 2F?N `;j s(y ;y 2 e?(`?n)" N e?"n ) k'k h N e?"n :

19 8 Estimate #4. For ` < N, k(p N ')`;j k h C K j'j + 4C e C N k'k h : Proof. Writing y i = F?N x i for x ; x 2 2 `;j, we have k(p N def ')`;j k h = ess sup X 'y? x ;x 2 2`;j 'y 2 JF N y (***) X br For each inverse branch, 'y? 'y 2 JF N y br ess sup y ;y 2 2F?N `;j JF N y 2 j'y? 'y 2 j JF N y JF N?s(x ;x 2 ) y 2 'y? 'y 2 JF N y JF N?s(y ;y 2 ) y 2 + j'y 2 j? JF N y JF N y 2! e?`" N j'y? 'y 2 j JF N y + j'y 2j JF N y 2 C s(f N?`y ;F N?`y 2 ) : (In the last line we have used the fact that F N?`y i 2 0 ; JF N y i = JF (F N?`? y i ), and the distortion estimate for JF.) We may now write ( ) (3) + (4) where and (3) := X br JF N I F?N `;j (4) := C X br JF N I F?N `;j ess sup y ;y 2 2F?N `;j 'IF?N `;j j'y? 'y 2 j s(y ;y 2 ) N : Observe that (3) is exactly the line above () in Estimate #2, and (4) diers only by a constant from the right side of the inequality in () in the same estimate. Putting these 4 estimates together and recalling that e?"n > N, we conclude that kp N 'k ( + C )K j'j + e?"n k'k + 0C e C N k'k h Kj'j + e?"n k'k! N for some K completing the proof of Lemma 3. Proof of Corollary to Lemma 3. To prove that the spectrum of P lies in the closed unit disk, it suces to show sup kp n k < : n

20 9 Using the fact that jp 'j j'j, we have for all k 2 + and ' 2 X, kp kn 'k e?"n kp (k?)n 'k + KjP (k?)n 'j. e?"kn k'k + K X j e?"jn A j'j : Since j'j C 00 0 k'k (see 3.2), we have shown that kp kn 'k K 0 k'k for some K 0. For n = kn + i; i < N, we have kp n 'k K 0 kp i 'k K 0 kpk N k'k Approximation of P by an operator of nite rank. The aim of this subsection is to prove Lemma 4. Let M 0 be the partition of into `;j -components (i.e. M 0 = D in.3), and let M N = N _ 0 F?i M 0. For ' :! C, we let E N (') denote the expectation of ' wrt m on elements of M N. For k 2 +, we dene ' k := 'I [ operator Q k on X dened by `k `; similarly, ' >k := 'I [ `>k `. Consider the Q k (') = P N (E N (' k )): Since the number of components on each level is nite, it is evident that Q k has nite rank. We will show that kp N? Q k k < N 0 for all suciently large k. In the discussion to follow it is convenient to write (P N? Q k )(') = P N ( ) + P N (' >k ) where = ('? E N (')) k. Note that E N ( ) 0. Estimate #5. kp N k 0 k'k h. Proof. As before there are 4 cases to consider: ` N and ` < N; k k and k k h. The k k term for ` N is dealt with a little dierently than before: k(p N def )`;j k = j I F?N `;jj e?`" javg( j F?N `;j )j + ess sup y ;y 2 2F?N `;j j y? y 2 j: Here, Avg( j F?N `;j ) = 0 and j y? y 2 j = j'y? 'y 2 j assuming `? N k (otherwise there is nothing to prove). This gives k(p N )`;j k N e?"n k'k h. The other 3 cases follow closely their counterparts in 3.4 except that the j j - terms are absent because E N ( ) 0. Note that as expected, k'k does not appear in the estimate.

21 20 Estimate #6. 9" k with " k! 0 as k! s.t. kp N (' >k )k (e?"n + " k )k'k + 0 k'k h: Proof. The estimates for ` N are identical with those in Estimates # and 3 and we omit them. Consider the k k -norm for ` < N. Let br >k denote the sum over all inverse branches with `br > k. Then k(p N (' >k )`;j )k X br X br >k e C JF N I F?N `;j >k e C m(f?n `;j ) m(`;j ) m(`;j ) k'k X `>k 'IF?N `;j m(`)e`" k'k e`br" which is < " 0 kk'k for some " 0 k with "0 k! 0 as k!. The k k h -norm for ` < N is dealt with as in Estimate #4; part of it refers back to the above estimate. Choosing k s.t. " k + e?"n < N 0 and remembering that e?"n >, we have 2 proved that for all ' 2 X, k(p N? Q k )'k < 5 k'k h + N 0 k'k N 0 k'k: We remark that Lemmas 3 and 4 are valid even when the number of `;j 's on each level is not assumed to be nite for ` N. (For ` < N, this niteness is used in a rather essential way in Estimate #2.) For ` N, it is used only to ensure that Q k has nite rank. This can be modied as follows. For each k, dene ' k :?'I k where k is a union of nitely many components of `;j chosen in such a way that X `;j6k m(`;j )e`"! 0 as k! : We may then dene ' >k := '? ' k and proceed as before Ruling out other eigenvalues of modulus. Let B be the -algebra of Borel subsets of. Recall that (F; ) is called exact if \ n0 F?n B is trivial in the sense that it contains only sets having -measure 0 or. We begin with the following observation:

22 2 Sublemma. Suppose that gcdfr(z)g =. Then for every `0 2 + ; 9t s.t. F t 0 ( 0 ) [ ``0 `. Proof. Because of the nature of Markov partitions, it suces to produce t 0 0 s.t. m( 0 \F?t 0 ) > 0 8t t 0 0, for then we could take t 0 = t 0 0 +`0. The existence of t 0 0 follows from the gcd assumption as in the proof of p (n) ij > 0 for irreducible aperiodic nite state Markov chains. Proof of Lemma 5. We prove the exactness of (F; ). Let A 2 \ n0 F?n B be s.t. (A) > 0. It suces to show that (A) >? " for every " > 0. From the Sublemma it follows that 9t = t (" ) 2 + and = (" ; t ) > 0 s.t. if B 2 B satises m( 0? B) <, then (f t B) >? ". We claim that it suces to show m( 0? F n A) < for some n > 0. Assuming this, we have, since A 2 F?(n+t ) B, that A = F?(n+t ) A 0 for some A 0 2 B. Hence (A) = (A 0 ) = (F t (F n A)) >? ". To produce an n with the property above, let `;j be s.t. m(a \ `;j ) > 0, and consider the increasing -algebra on `;j dened by M k ; k = ; 2; : : : (see 3.5 for denition). Clearly, E(I A j M k )! I A m-a.s. as k!. Pick a typical point x 2 A \ `;j and choose a suciently large n s.t. F n (M n (x)) = 0. Then our distortion estimate for m (see Lemma 2) gives mf n (A \ M n (x)) m( 0 ) m(a \ M n(x)) m(m n (x)) as required. Proof of Proposition A(2). We nish by explaining how the exactness of (F; ) implies the conclusion of Proposition A(2). Let ' 2 X; ' 6 0, be s.t. P (') = ' for some 2 C with jj =. We write ' =, which is legitimate since c 0 > 0 (Lemma 2), and observe that 2 L 2 (m) because j`;j j k'k c 0 e "` and ` e 2`" m(`) <. With m uniformly equivalent to, this puts 2 L 2 (). Consider U : L 2 ()! L 2 () dened by U( ) = F and let U be the adjoint of U. Then U () = R because (U ) R R d = (U )d = ( F )dm = d for all 2 L 2 (). It follows from this that = U(), R P ()dm = R P () hence = n U n () 8n, which implies that is measurable wrt F?n B 8n. The exactness of (F; ) then tells us that const a.e. This in turn forces = because P () = P () = (and 6= 0), proving that is the only spectral point of P with modulus. Let X be the image of the projection associated with 2 (P ). We have shown that dim X <. The number of Jordan blocks for P j X cannot exceed because is the unique invariant density of F, and there can be no subdiagonal 's in the block because that would contradict sup n kp n k <.

23 22 We remark that there are other standard ways of dealing with eigenvalues on the unit circle. One could, for instance, argue that for positive operators like P, all eigenvalues of modulus are n th roots of unity, and for P their eigenfunctions are invariant densities of F n. The problem then boils down once again to proving the ergodicity of (F n ; ) for all n as we have shown. 4. Exponential Decay of Correlations Recall that we have constructed a Markov extension F : (; ~) over our dynamical system of interest f : (M; ) where is an SRB measure and :! M sends ~ to. We have also constructed a quotient system F : (; ) by collapsing s -leaves in F : (; ~), and the projection map :! sends ~ to. Let us use the following convention: if ' is a function on M, then ~' is the lift of ' to, i.e. ~' = ' ; and if ~' is constant on s -leaves, then we will confuse it with the function on called '. (It would have been logical to write ~ F : ( ~ ; ~) instead of F : (; ~) but let us not do that.) We assume for the rest of Part I that (f n ; ) is ergodic for all n. 4.. Reduction of problem. The purpose of this subsection is to reformulate the problem of decay of correlations for (f; ) in a way that the properties of the Perron-Frobenius operator studied in Section 3 can be brought to bear. Recall that H := f' : M! R j 9C = C ' s.t. 8x; y 2 M; j'x? 'yj Cd(x; y) g: We will use D n ('; ; ) to denote the correlation between ' and f n wrt the probability measure, i.e. D n ('; ; ) := ( f n )'d? 'd d : Our rst observation is that to prove Theorem 2, it suces to prove the corresponding result for F : (; ~) and test functions of the form ~'; ~ with '; 2 H. This is because D n ('; ; ) = D n ( ~'; ~ ; ~), which is straightforward to verify. Next we observe that we may assume gcd fr(x) : x 2 0 g =. Suppose not. Let N = gcdfrg. Instead of F : (; ~), we will consider F N :? (N ) ; ~ (N ) where (N ) := [ k=0 kn and ~ (N ) := ~ j (N ) normalized. Since ~ (N ) is an f N -invariant probability measure on M and it is absolutely continuous wrt, it follows from the ergodicity of (f N ; ) that ~ (N ) =. Suppose we have the desired result for F N :? (N ) ; ~ (N ) and hence for f N : (M; ). For given '; 2 H, exponential decay of D nn ('; f i ; ) for i = 0; ; : : : ; N? clearly implies that of D n ('; ; ). From now on we assume gcd frg =. The following geometric fact about the \sizes" of the `;j 's is used to relate the Holder property of functions in H to a corresponding property for their lifts to :

24 23 Sublemma. 8x 2 ; diam(f k (M 2k (x))) 2C k. Proof. Let y ; y 2 2 M 2k (x). Then 9^y 2 u (y ) \ s (y 2 ). Suppose M 2k (x) `. Then F?`^y; F?`y 2 are both in 0 and they lie on the same s -leaf. By (P3), we have d(f k ^y; F k y 2 ) < C`+k. Applying (P4)(a) to ^y and y and noting that s(f k ^y; F k y ) k, we obtain that d(f k ^y; F k y ) < C k. We now proceed to argue that D n ( ~'; ~ ; ~) can be approximated by a quantity that involves only F : (; ) and functions on, and that the error decreases exponentially with n. For each n 2 +, let k = k(n) be a number < n to be 2 determined. Approximation #. Dene k on (or ) by k j A = inff (x) : x 2 F k Ag for every A 2 M 2k. Then for some C 0 = C 0 ('; ). jd n?k ( ~'; ~ F k ; ~)? D n?k ( ~'; k ); ~j C 0 k Proof. It follows from the Sublemma that j ~ F k? k j C (2C k ). Hence the quantity in question is ( ~ F k? k ) F n?k ~'d~ 2C (2C k ) max j'j C 0 k : + ( ~ F k? k )d~ The next 2 steps are made slightly more complicated by the fact that F need not be one-to-one. Approximation #2. Let k be as above, and let ' k be dened analogously. Let ' k ~ denote the signed measure whose density wrt ~ is ' k, and let ~' k := d(f k (' k ~))=d~. Then Dn?k ( ~'; k ; ~)? D n?k ( ~' k ; k ; ~) C 00 k for some C 00 = C 00 ('; ). ~'d~ Proof. The quantity in question is ( k F n?k )( ~'? ~' k )d~ (2 max j j) j ~'? ~' k jd~: + kd~ ( ~'? ~' k )d~

25 24 Letting j j denote the total variation of a signed measure, and noting that F k (( ~' F k )~) = ~'~, we have j ~'? ~' k jd~ = j ~'~? ~' k ~j = F k (( ~' F k )~)? F k (' k ~) ( ~' F k? ' k )~ = j ~' F k? ' k jd~; and this last quantity has been estimated above. Finally, we verify that D n?k ( ~' k ; k ; ~) can be expressed purely in terms of objects related only to F : (; ). First, ( k F n?k ) ~' k d~ = kd(f n?k ( ~' k ~)) = kd(f n (' k ~)); and since k is constant on s and F commutes with, we have kd(f n (' k ~)) = kd( F n (' k ~)) = which, in the language of Section 3, is equal to kp n (' k )dm: kd(f n (' k )); Also, ~' k d~ kd~ = d(f k (' k ~)) kd = ' k d kd: 4.2. Estimating D n?k ( ~' k ; k ; ~) and completing the proof. >From the last subsection we have D n?k ( ~' k ; k ; ~) = k P n (' k )? max j j P n (' k )? max j j C 00 0 where k k is the norm introduced in 3.2. P n (' k )? kdm ' k dm ' k dm dm

26 25 Sublemma. 9C ' depending only on ' s.t. kp 2k (' k )k C ' for all k. Proof. On each `;j ; = P 2k can be written as i i 2k where i 2k is the contribution from each of the inverse branches of F 2k. Since ' k is constant on elements of M 2k ; P 2k (' k ) on each `;j has the form c i i 2k, and i jp 2k (' k )(x)? P 2k (' k )(y)j X i jc i jj i 2k(x)? i 2k(y)j max j'j (x) C s(x;y) as in the proof of lemma 2. Let > supfjj : 2 (P ); 6= g. The sublemma above suggests that we write P n (' k )? ' k dm = P n?2k (P 2k (' k ))? P 2k (' k )dm : Since for h 2?R X; hdm is the projection of h onto the? d subspace spanned by, the quantity above is C n?2k P 2k (' k )? const n?2k : P 2k (' k )dm Combining the arguments in the last two subsections, we see that if for instance we let k = n; 2? 0; 2, then we have for all '; 2 H, D n ('; ; ) C n for = maxf ; (?2) g. 5. Central Limit Theorem 5.. CLT for dynamical systems: background information. In this subsection we review 3 known results related to the asymptotic normality of random variables generated from dynamical systems. Notations in 5. are independent of those in the rest of this paper. A. Conditions for CLT for measure-preserving transformations: a theorem of Gordin.

27 26 Theorem [G]. Invertible case. Let T : (; M; ) be an invertible measure preserving transformation of a probability space, and let ' 2 L 2 () be s.t. E' = 0. Suppose there exists a sub--algebra M 0 M s.t. T? M 0 M 0 and Then where X j0 je(' j T?j M 0 )j 2 + X j0 X n? p ' T i n i=0 2 = lim n! n n? je(' j T j M 0 )? 'j 2 < distr?! N (0; ) X i=0 ' T i! 2 d: Noninvertible case. If T is not invertible, take M 0 = M and disregard the second term in (). Otherwise same hypotheses and same conclusion. Remark. Suppose for instance that (T; ) is a K-automorphism and M 0 is such that T j M 0 " M and T?j M 0 # the trivial -algebra as j!. Then () could be viewed as an L 2 -version of the K-property seen through the eyes of the random variable '. B. Condition for CLT and the Perron-Frobenius operator. Exponential decay of correlations (as dened in.4) alone does not imply (), but if T : is noninvertible and admits some additional structures so that a Perron-Frobenius operator P can be dened, then () can be expressed in terms of P. In particular, assuming that all relevant functions belong in a suitable space, then a gap in the spectrum of P implies that je(' j T?j M)j 2 is a geometric series. Following [Ke] and putting j0 what is there into a slightly broader context, we make this connection precise: Let T : (; M; ) be noninvertible, and suppose that on (; M) there is a reference measure m with respect to which T is nonsingular and = m for some with c > 0. Let P (') be the density wrt m of the measure T ('m), and let U : L 2 () be the operator dened by U(') = 'T. It is straightforward to verify that for all ' 2 L 2 (): U j (') = P j (')= U j U j (') = E(' j T?j M) where U is the adjoint of U: Using these, one sees immediately that je(' j T?j M)j 2 d = = ju j U j 'j 2 d = ju j 'j 2 d j(u j U j ') 'd j'j = j'j jp j (')jdm: ju j 'j T j d ():

28 27 C. Positivity of the variance. The following fact is communicated to me by Bill Parry (see also [PP]). Lemma. Let T : (; M; ) be an mpt and let ' 2 L 2 () be s.t. R 'd = 0. Suppose also that R ('T n )'d! 0 exponentially fast. Then n i ' = T? for some 2 L 2 (). R n? 0 ' T i 2 d! Proof of Theorem 3. We now return to the setting and notations prior to 5.. Let ' 2 H. Again we lift to ~' :! R and note that a CLT for f ~'F i g implies one for f'f i g. Also, once the convergence in distribution is proved, the positivity of 2 follows automatically from Theorem 2 and the lemma above. Observe, however, that technically we could not appeal to 5.A, B and nish immediately, for our Perron-Frobenius operator is associated with F : while ~' is dened on. Here is one way to reconcile these dierences: Let B be the Borel -algebra on. Let B 0 := f? A : A 2 Bg, and let ' 0 := E ~ ( ~' j B 0 ). Claim #: It suces to show j0 je (' 0 j F?j B)j 2 < where j j 2 is wrt. Claim #2: The sum above is nite. To verify Claim #, let ^F : ( ^; ^) denote the natural extension of F : (; ~). Then ^ is homeomorphic to ft; x 0 ; x ; : : : ) 2 (=? u ) N : F x i+ = x i g where =? u is the quotient of obtained by collapsing along u -leaves. Let ^B 0 be the -algebra ^?? B 0 where ^ : ^! is the natural projection. Then ^F ^B 0 ^B 0. Let ^' : ^! R be the lift of ~'. Noting that a CLT for f ^' ^F i g is equivalent to one for f ~' F i g, we proceed to verify Gordin's condition () for the conditional expectations ^' j := E^ ( ^' j ^F j ^B 0 ). For j 0; ^F j ( ^B 0 ) is generated by sets of the form f(t; x 0 ; x ; : : : ) : t 2 F j? fag; x 0 = F j a; : : : ; x j = ag; a 2 : Thus j ^' j? ^'j < C j and so j0 j ^' j? ^'j 2 <. As for ^'?j ; j 0, since the order of conditioning is immaterial, ^'?j can also be written as E (' 0 j F?j B), and so it suces to prove j0 je (' 0 j F?j B)j 2 < as claimed. We are now in the situation of 5.B. If we know ' 0 2 X, then, recalling that R '0 dm = 0, we could nish by saying jp j (' 0 )jdm C 00 0 kp j (' 0 )k C j k' 0 k: Now ' 0 2 X if ' 0 2 X (this uses the fact that j ` = F?`). To prove Claim #2, it remains only to show