Countable Markov Partitions Suitable for Thermodynamic Formalism

We study hyperbolic attractors of some dynamical systems with apriori given countable Markov partitions. Assuming that contraction is stronger than expansion we construct new Markov rectangles such that their crossections by unstable manifolds are Cantor sets of positive Lebesgue measure. Using new Markov partitions we develop thermodynamical formalism and prove exponential decay of correlations and related properties for certain H\"older functions. The results are based on the methods developed by Sarig.


Introduction
Examples of one-dimensional maps with countable Markov partitions go back to the Gauss transformation, and further developments appeared in particular in [24], [3], [31]. Beginning in [4], theorems about ergodic properties of such maps are often referred to as Folklore.
More recently an interest in such maps was motivated by works on ergodic and statistical properties of quadratic-like and Hénon-like maps. The study of such maps is typically based on various tower constructions, see in particular [16], [17], [7], [8], [32]. Power maps defined on the tower satisfy hyperbolicity and distortion estimates.
Although the following remark is not directly related to the results of our paper, it might be useful for the further work in the area under discussion. Remark 1.1 Since numerical evidence for the existence of a strange attractor was presented in the original Hénon paper [13], rigorous results were only obtained in unspecified small neighborhoods of one-dimensional maps. One possible approach to that problem is to prove that in a sufficiently small neighborhood of the classical Hénon values, a H = 1.4, b H = .3, there is a positive Lebesgue measure set M, such that for (a, b) ∈ M, Hénon maps f a,b have SRB measures with strong mixing properties. The main difficulty in that direction is to design a set of checkable numeric estimates which can be maintained through the induction. In the one-dimensional case such estimates were used in [22], [15], [11].
We study apriori given two-dimensional systems with countable Markov partitions satisfying hyperbolicity and distortion conditions. In [18] we proved strong mixing properties of such systems assuming distortion condition D2, requiring boundedness of the quotient of the second derivatives over the first derivative. This condition is too strong for our purposes since it does not hold for power maps induced by quadratic and Hénon maps.
Here we return to the more general setting of [20], [21] where only boundedness of the quotient of the second derivatives over the square of first derivatives is assumed. In order to study the decay of correlations we require additionally that contraction in our models grows faster than expansion, see condition H5 below. Condition H5 naturally holds for power maps generated by Hénon-like maps with Jacobian less than 1. We also require distortions of our initial maps to be uniformly bounded, see condition BIV below. That is a standard requirement for maps defined on a tower. The main new idea of the paper is to develop thermodynamic formalism by using special Markov rectangles such that their intersections with unstable manifolds are Cantor sets of positive Lebesgue measure.
2 Description of models and statement of results. 1. Description of the model.
The setting for our model is the same as in [18] - [21]. To summarize, let Q be the unit square. Let ξ = {E 1 , E 2 , . . .} be a countable collection of fullheight, closed, curvilinear rectangles in Q. Hyperbolicity conditions that we will recall below imply that the left and right boundaries of E i are graphs of smooth functions x (i) (y) with dx (i) dy ≤ α for 0 < α < 1. Assume that each E i lies inside a domain of definition of a C 2 diffeomorphism F which maps E i onto its image S i ⊂ Q. The images F| E i (E i ) = f i (E i ) = S i are disjoint, full-width strips of Q which are bounded from above and below by the graphs of smooth functions y i (x), | dy (i) dx | ≤ α. We recall geometric and hyperbolicity conditions from [21].

Geometric conditions.
For z ∈ Q, let ℓ z be the horizontal line through z. We define δ z ( We recall some notation.
Given a finite string i 0 . . . i n−1 , n ≥ 1, we define inductively We also define curvilinear rectangles R i −m ...i −1 ,i 0 ...i n−1 by If there are no negative indices then respective rectangle is full height in Q.
The following is a well known fact in hyperbolic theory, see [21].
Proposition 2.1 Any C 1 map F satisfying the above geometric conditions G1-G3 and hyperbolicity conditions H1-H4 has a "topological attractor"

The infinite intersections
define C 1 curves y(x), |dy/dx| ≤ α, which are the unstable manifolds for the points of the attractor.

The infinite intersections
define C 1 curves x(y), |dx/dy| ≤ α, which are the stable manifolds for the points of the attractor.

The infinite intersections
define points of the attractor.

Distortion condition.
We formulate certain assumptions on the second derivatives. We use the distance function d((x, y), (x 1 , Our first condition is the same as in [21]; we recall it below. Let f i (x, y) = ( f i1 (x, y), f i2 (x, y)). We use f i jx , f i jy , f i jxx , f i jxy , etc. to denote partial derivatives of f i j , j = 1, 2.
We define We assume that there exists a constant C 0 > 0 such that the following Distortion Condition holds:

5.
A result about systems satisfying geometric, hyperbolicity and distortion conditions: Our conditions imply the following Theorem proved in [20] and [21].
Theorem 2.2 Let F be a piecewise smooth mapping as above satisfying the geometric conditions G1-G3, the hyperbolicity conditions H1-H4, and the distortion condition D1.
Then, F has an SRB measure µ supported on Λ whose basin has full Lebesgue measure in Q. The dynamical system (F, µ) satisfies the following properties.
(b) F has finite entropy with respect to the measure µ, and the entropy formula holds where D u F(z) is the norm of the derivative of F in the unstable direction at z.
where the latter limit exists for Lebesgue almost all z and is independent of such z.
6. Additional distortion and hyperbolicity conditions and statement of the main theorems.
(a) Properties of the function φ (z) = − log(D u F(z)) are important when applying thermodynamic formalism to hyperbolic attractors. We consider systems satisfying conditions of Theorem 2.2 and some extra hyperbolicity conditions, which can be used for power maps arising from Henon-type diffeomorphisms.
We explore a general principle that can be stated as : Contraction increases faster than expansion -see hyperbolicity condition H5 below. For such systems we construct new Markov partitions such that the pullback of φ (z) into a respective symbolic space is a locally Hölder function.
New Markov rectangles are Cantor sets, such that their one-dimensional crossections by W u (z) have positive Lebesgue measure.
(b) We consider a class of system which satisfy conditions of the Theorem 2.2 as well as the following additional assumptions.
i. Bounded Initial Variation.
BIV. There exists B 0 > 0 such that for all i and all BIV does not allow unbounded oscillations of widths for initial rectangles; it implies that a = b in condition G3: ii. Contraction grows faster than expansion. We assume that there is a constant a 1 satisfying where a is from (6), such that for each j, for each z ∈ E j , and for any vector v in the stable cone K s Condition H5 means that up to a uniform factor, contractions of f j grow faster than expansions. In particular it implies that up to a uniform factor, heights of S i ∩ E k are smaller than widths of E k for all k ≤ i.

Remark 2.3
Uniform hyperbolicity and distortion conditions were used in [20], [21] to extend the classical approach of [6] and to study ergodic properties of systems with countable Markov partitions. By combining D1 and H5 we can add methods of thermodynamic formalism.
(c) Let H γ be the space of functions on Q satisfying Hölder property with We state our main Theorems.

Theorem 2.4 (Exponential Decay of Correlations)
Let F be a piecewise smooth mapping as above, satisfying geometric conditions G1-G3, hyperbolicity conditions H1-H5, distortion condition D1, and the BIV condition. Then the system (F, µ) has exponential decay of correlations for f ∈ H γ and g ∈ L ∞ (µ). Namely there exists Theorem 2.5 (Central Limit Theorem) Let (F, µ) satisfy the assumptions of Theorem 2.4 and suppose that f ∈ H γ . If f dµ = 0 and f cannot be expressed as g − g • F for g continuous, then there is a positive constant σ = σ ( f ), such that for every t ∈ R, 3 Hölder properties of log(D u F(z)) and Markov partitions in the phase space.
1. The key step toward the proof of Theorem 2.4 is to establish that the pullback of the function log D u F into the respective symbolic space is Hölder continuous. Then one can follow the Ruelle-Bowen approach ( [25], [9]), in particular using results of Sarig [26] - [28] to develop thermodynamic formalism for the systems under consideration. Hölder properties of the pullback of log D u F into symbolic space follow from Hölder properties of log D u F in the phase space.
In order to get an appropriate symbolic space, we construct a partition of a subset of positive measure C ⊂ Λ, such that the first return map to C is Markov. Elements of the Markov partition of Λ are elements of the Markov partition of C and their orbits before the first return. Elements C i of the Markov partition C have the following property. For is a Cantor set of positive linear Lebesgue measure.

Cantor sets that we construct are inscribed in curvilinear rectangles
We define the height of Hyperbolicity conditions imply that stable boundaries of rectangles belong to stable cones. Since standard horizontal lines belong to unstable cones, and stable and unstable cones are separated, we get the following.
For every ε 1 there is an ε 0 such that if then for all m ≥ 1, n ≥ 1, the ratio of lengths of any two unstable curves Similarly it follows from hy-perbolicity conditions, that if l is the length of a standard horizontal crossec- 3. Admissible objects.
We define the following strings of indices as admissible: is admissible, then all rectangles obtained by moving the comma in the index to the left or to the right are admissible. A one-sided sequence In particular all strings [i j], i ≥ j are admissible, and thus, the respective rectangles are admissible.
Note that distortion estimates may be satisfied on non-admissible rectangles if their heights are small enough, but we ignore that possibility, and organize our construction based on condition A1. 4. We estimate the variation of log D u F on admissible two-dimensional curvi- For any function a(x, y) the variation of a(x, y) over a rectangle R is defined as The function log D u F is locally Hölder on admissible rectangles for some C > 0, θ 0 < 1.
Note that on initial rectangles E i estimate (13) is satisfied because of BIV.
5. The proof of the following Proposition is similar to the proof of Proposition 5.1 in [18]. (13) with some C and θ 0 independent of m, n and determined by hyperbolicity and distortion conditions.
Note that in [18] and [19] we proved Hölder property of log D u F on arbitrary Here we prove it for admissible rectangles. Proof.
We consider two points z 3 , z 4 such that W s (z 3 ) = W s (z 4 ) and for which we can connect z 1 to z 3 and z 2 to z 4 along their respective unstable manifolds. We define the following Now we bound each term on the right hand side of the inequality Estimates of |logD u F(z 1 )−logD u F(z 3 )| and |logD u F(z 4 )−logD u F(z 2 )| are the same as estimates (15) − (28) in the proof of Proposition 5.1 in [18]. Then we get Similarly, (b) The second part of the proof, depending on m, follows again the ideas in [18] and [19] but also utilizes condition H5. We are left with estimating the difference Note that the BIV condition implies that the above difference is uniformly bounded on full-height rectangles. From [21] we get that the hyperbolicity conditions imply that any unit vector in K u α at a point z ∈ E i , in particular a tangent vector to W u (z), has coordinates (1, a z ) with |a z | < α. Thus we need to estimate Now we are moving along γ 3 ⊂ W s (z 3 ) connecting z 3 and z 4 . We cover γ 3 by rectanglesR k for which the widths ∆x and lengths ∆y satisfy |∆x| < α|∆y|. As in [18], we get (18) by estimating differences forz,z ′ ∈R k ∩W s (z 3 ), and To estimate (19) we use the mean value theorem and get on each rect-angleR k an estimate not exceeding Let The sum of the contributions from (21) is bounded by Const sup Since the distortion condition D1 is expressed using the width of E i 0 , we use condition H5.
On an admissible rectangle Let h max be the maximal height of S i −m ...i −1 and let w min be the minimal width of E i 0 . From condition H5 we know that contraction of f i is stronger than a i 1 . Since the rectangle is admissible, the sum of the indices satisfies i −m + . . . i −1 ≥ i 0 . Since contraction of the composition is stronger than a i 0 1 , it follows that As each index is at least 1 we get that i 0 ≥ m and thus Therefore the heights of rectangles R i −m ...i −1 ,i 0 decay exponentially comparatively to the width of E i 0 . Since |γ 3 | < h max and w min < δ z (E i 0 ) for any z ∈ E i 0 , we can apply D1 and obtain the following bound for the sum of the contributions from (19), We estimate (20) as in [18], [19]. We assume by induction As in [19] one can assume by taking if needed instead of F some power of F 1 Note that differently from [19] the variation of log D u F along stable manifolds inside admissible rectangles is controlled not by using bounded distortions of inverse maps, but from (24) and (27).
We define full height Cantor sets C n inside full height rectangles E n , n ≥ 1 by where i 0 = n, k ≥ 1 and all [i 0 . . . i k−1 ] are admissible strings.
As an example let us consider several rectangles with indices starting from 1. It follows from the definition that [11] is admissible and [1i], i > 1 are not. Then R [11] is the only defining rectangle of order two for the As each index is at least 1 we get from the definition of admissible rectangles that inadmissible indices satisfy i N > N. Geometric condition G3 implies that on any unstable manifold, the relative measure of the union of rectangles with indices greater than N decays exponentially. Then uniformly bounded distortion implies that the total relative linear measure of gaps in an unstable manifold of any defining rectangle of order N has uniform exponential decay.
This implies the following Corollary.
Corollary 3.2 There is a c 0 > 0 such that for any initial full height rectangle E n and respective full height Cantor set C n constructed inside E n and for any z ∈ C n ∩ Λ the relative linear measure of C n in W u (z, E n ) is greater than c 0 . Moreover that relative measure tends to one when n → ∞.
Also as in [18], Remark 5.10, we get the following Corollary from distortion estimates.
As Corollary 3.3 is valid for all defining rectangles we get Corollary 3.4 Let C n be the full height Cantor set constructed inside E n , let z ∈ C n ∩ Λ, and let | W u (z,C n ) | be the linear Lebesgue measure of C n in W u (z, E n ). Then for any two points z 1 , z 2 ∈ C n ∩ Λ, it holds that For example gaps of C 3 are: E 311k , k > 5 -gaps of order 3, and so on.
The following example illustrates Markov relations between Cantor sets C i .
Consider an admissible rectangle E 112 . Then F(E 112 ) = R 1,12 is a subrectangle of a non-admissible rectangle E 12 . However, because of H5, F 2 (E 112 ) = R 11,2 has height less than the width of E 2 .
We will use the notation and in general, We have that As the sum of indices in [112] is greater than 2, the inclusion in (35) is not an equality. Namely strings [23] and [24] are not admissible, but [1123] and [1124] are admissible. The image F 2 (C 112 ) covers respective slice of C 2 and also covers some parts of gaps C 23 and C 24 used in the construction of the Cantor set C 2 .
Consider the union of all full height Cantor sets C n : For z ∈ C i 1 ...i n let us denote W u (z, Let T be the first return map to C generated by F . Similarly to (35) above we get the following Markov properties.
Let T 1 be the first return map to C 1 ⊂ C . Next we estimate the measure of points in C 1 which return to C 1 after at least n iterates of F. Proposition 3.6 Let B n be the set of points in the domain of T 1 such that the return time for z ∈ B n is greater than n. For some C > 0 and 0 < β < 1, We begin by proving (39) for the first return map T onto C .
. . x n x n+1 . . .) / ∈ C , and F n x ∈ C is the first return.
As y / ∈ C , there is a coordinate y k such that y k = x k+1 > y 0 + . . . + y k−1 = x 1 + . . . + x k . If k ≥ n, then Suppose k < n satisfies Then x k+1 ≥ k + 1 because each coordinates is at least 1.
As all images F k x, 1 ≤ k < n do not belong to C , there is a coordinate If N ≥ n, then and we get a contradiction as above to F n x ∈ C . So N < n and we get Proceeding as above we get x n ≥ n.
As widths of E i decay exponentially, the measure of the collection of x satisfying (41) decays exponentially. Since the measure of C is positive, we get that the measure of points which do not return to C after n iterates is less than for some C 1 > 0 and 0 < β 1 < 1.
Next we note that if z ∈ C 1 is mapped by T into C i , then because all transitions from C i to C 1 are admissible, z will be mapped into C 1 by the next iterate of F. Therefore points which do not return into C 1 after n iterates are subdivided into two subsets: points which did not return into C after n iterates and points which returned into C i , i > 1, after n − 1 iterates and at the next iterate were not mapped into C 1 . Because of uniformly bounded distortion the measure of the second set is less than where 0 < γ 0 < 1. Then (39) follows from (42) and (43) if we take C = 2C 1 and β = max{β 1 , 1 − γ 0 }.
This concludes the proof of Proposition 3.6.

First return maps and respective transition matrices.
Note that on every unstable leaf, relative measures of C i inside E i are uniformly bounded away from 0. Together with uniformly bounded distortion it implies that in the orbits of the first return map each gap is substituted by a union of new gaps and a Cantor set of relative measure greater than some uniform c 0 > 0. Then at the end of that construction we get, up to a set of measure zero, Note that C ik can be unions of several Cantor sets which belong to disjoint full height rectangles. For example consider C 1113 ⊂ E 1113 . Admissible rectangle E 1113 is mapped as follows, As E 113 , E 13 are inadmissible, we get that T = F 3 maps C 1113 onto C 3 in a Markov way. Similarly T = F 3 maps C 1123 onto C 3 in a Markov way. The correct labeling is provided by respective strings of the original alphabet starting width i and ending with k.
To get an authentic Markov partition which generates a transition matrix of 0's and 1's we partition each C j 0 j 1 into subsets As in the proof of (39) we get that the length of the above strings starting from j 0 and ending with j 1 is at most j 1 − j 0 + 2.
The union of C j 0 i 1 ...i n−2 j 1 forms a Markov partition of C . Using Proposition 3.5 we get Lemma 3.7 To any one-sided T -admissible sequence of transitions corresponds a unique one-sided sequence of the original alphabet such that Up to a set of µ measure zero, any point of the attractor is uniquely labeled by a two-sided sequence of admissible transitions We consider a new alphabet Ω corresponding to the elements of the tower from Lemma 3.7 and get a subshift Recall that a subshift is topologically mixing if for any states a and b there is n(a, b) such that for n ≥ n(a, b) there is an admissible word of length n starting from a ending with b.
We will need the following Proposition.
Note that although our original map is clearly topologically mixing elements of the Markov partition are Cantor sets, so the statement is not obvious.
Proof of Proposition 3.8. By construction any Cantor set C which coincides with some element of the tower is mapped by some iterate of F onto a full width substrip of some Cantor set C i . As the image of any C i (including C 1 ) contains a full with substrip of C 1 , we get that all consecutive images of C i have Markov intersections with C 1 .
It remains to prove that for any element ∆ of the tower there is an n(∆) such that F n C 1 has Markov intersection with ∆ for n > n(∆) . By construction any ∆ = F k(∆) P where P is a full height Cantor subrectangle of some C i . So it is enough to prove that F n C 1 intersects C i for n > n(i). But C 1 contains a full height Cantor subset C 11...1i and all images of C 1 have Markov intersections with C 11...i . That proves Proposition 3.8.
Next we consider the first return map T 1 induced by T on C 1 . Consider the Markov partition M P 1 of C 1 generated on C 1 by T 1 . By construction T 1 maps its domains (which are full height Cantor sets) onto full width substrips of C 1 . Therefore the transition matrix corresponding to the map T 1 on M P 1 consists of all 1's.
Remark: [1] will correspond to our choice of state [a] in the symbolic dynamics in later sections.
By first reducing to functions defined on one-sided sequences, we will show that our transfer operator (54) has the spectral gap property (57) on a particular Banach Space. This property implies exponential decay of correlations and Central Limit Theorem for functions defined on one-sided sequences. Then we can extend these results to certain functions defined on two-sided sequences, Theorems 2.4 and 2.5, respectively. This exchange between the two settings is a consequence of the reduction from two-sided shifts to onesided shifts following from the classical arguments of Ruelle and Bowen, see [25], [9]. In the case of an infinite alphabet, detailed reduction arguments can be found in sections 4 and 5 of [33].
If we restrict our consideration to Hölder functions on Q, then we are left with the proof of the spectral gap property (SGP) of the transfer operator acting on a suitable space L of functions defined on one-sided sequences of the alphabet Ω. We do this in the next sections, following [10] and [28].
2. Thermodynamic Formalism. Now by following [25], [9] we develop thermodynamic formalism on the space of one-sided sequences for the function Φ(x, y) = −log|D u F|.
Let E i ⊂ E i be an element of the Markov partition M P. For each E i we fix some unstable manifold W u 0 , and to any (x, y) ∈ E i we let correspond (x, y 0 ) = W s (x, y) ∩W u 0 . We define Then we can construct a Hölder function on one-sided sequences cohomologous to Φ(x, y) in the following way, We call φ the potential.
The transfer operator, L φ is defined as In the next several sections we consider the space of functions on one-sided sequences for which we can prove the spectral gap property for L φ . We denote by X the space of one-sided sequences. From this point forward, points x, y will be one-sided sequences belonging to X .

Induced system.
Just as in general case, we get SGP as a result of a particular induction procedure, see [10], [28], and references to earlier works in [28]. In our setting we induce on C 1 ; here our [a] is [1]. Let φ := We call φ : X → R the induced potential.

Gurevich Pressure.
We introduce a few preliminary definitions and results.
Then the limit, called the Gurevich Pressure, exists, see [26], and we can calculate it explicitly in our setting.
Let T 1 be the first return map to C 1 . Each periodic orbit of T 1 is contained in some admissible cylinder of F, also periodic but which has, in general, a larger period. Moreover there are F-strings of arbitrary large periods which correspond to a given T 1 period. Admissible cylinders of the same T 1 periods but with different T 1 labels do not intersect.
Proposition 3.1 and uniformly bounded distortions of D u F imply that the contribution to P G (φ ) from each periodic T 1 orbit differs from the length of the horizontal cross section of the respective two-dimensional cylinder by a uniformly bounded factor. That implies that the quantities Z n (φ ) are uniformly bounded from above.
As the measure of the Cantor set C 1 is positive, we get that Z n (φ ) are uniformly bounded from below. Thus, Remark: This calculation works for any C i , so as in the general case (see [28]) the Gurevich pressure is independent of the choice of partition set [a].

Spectral gap.
Following [10], [28] we would like to show that L φ has spectral gap on the appropriate Banach space L , defined below in (61). The implications of such a property are as follows.
If L φ has spectral gap then it can be written as and the spectral radius, ρ, of N is less than λ . Since ρ < λ , exponentially fast as n → ∞.
In our setting, λ = e P G (φ ) = 1 and where h is the eigenfunction of L φ and ν is the eigenmeasure of L * φ .
Following [10], [28] we introduce the a-discriminant The Discriminant Theorem 6.7, from [28], gives necessary and sufficient conditions for the spectral gap based on certain properties of the a-discriminant.
Specifically, it links the strict positivity of the discriminant to the spectral gap property. It involves the Gurevich pressure evaluated with respect to the induced system. We state one of the properties relevant to our setting.
Proposition 4.1 Let X be a topologically mixing TMS and suppose φ : X → R is a weakly Hölder continuous function such that P G (φ ) < ∞. If for some state a, then φ has the SGP on the Banach space L defined below in (61).

Recall that
We begin by calculating Z 1 (φ + p): Note that this β comes from the estimate in Proposition 3.6.
Next we calculate Z 2 (φ + p), Similarly we prove by induction for all n, Z n (φ + p) ≤ C n φ M n 1 , and thus, We combine the following properties from [28] with the fact that in our setting P G (φ ) = 0 for φ (x) cohomologous to Φ(x, y).
For our p > 0, For all x, y ∈ X , let t(x, y) = min{n : x n = y n } Let [1] be the collection of one-sided sequences, x = .
As proved in [10] there is a positive function h 0 : Z + → R with the following properties.
Consider the set of continuous functions { f : f L < ∞} where Then L is an L φ -invariant Banach space, and L φ on L is a bounded operator with spectral gap. Additionally the eigenfunction h of Ruelle operator belongs to L and for any bounded Hölder function ψ it holds that ψh ∈ L (62) Note that bounded Hölder functions belong to L .
8. It follows from Propositions 4.2 and 4.3 that the discriminant is strictly positive, and thus, by Proposition 4.1, L φ has spectral gap on the Banach Space L . As in [10], [28] this implies that (σ , µ φ ) has exponential decay of correlations.
Theorem 4.4 For σ a one-sided full shift, consider φ the potential defined in (53), and let µ φ be the respective invariant measure. Then (σ , µ φ ) has exponential decay of correlations for bounded Hölder functions f and g ∈ L ∞ (µ φ ). Namely there exists 0 < η 1 < 1 such that Note that f h ∈ L because of 62. As in [10], [28], the subsequent result follows. Remark: It follows from Theorem 1.1 part d in [10] that for g, bounded and Hölder continuous, P(φ + tg) is real analytic in a neighborhood of 0.
9. Exponential Decay of Correlations and Central Limit Theorem for functions of two variables.
One can follow arguments of Section 4 of [33] to reduce the estimate of the n-th correlation for functions defined on the two-sided shift to the following estimate for functions defined on the one-sided shift, where h dm = dν and ν is the eigenmeasure of L * φ as in (58). Here 2k < n, and f k is a piecewise constant approximation of f on cylinders of length k. Thus f k is a Hölder function bounded by max | f |. From (62) we get f k h ∈ L .
As in [33] Section 4 we get that norms L 2k φ ( f k h) L are uniformly bounded by a constant which only depends on max | f |. From (64) we get an estimate similar to (63) but with a different constant and a different 0 < η < 1. That proves Theorem 2.4. Theorem 2.5 follows from arguments in Section 5 of [33] regarding a result referred to as Theorem [G] from [12]. Using the spectral gap property, one obtains the following estimate, where, as above, f 0 is a bounded Hölder approximation of f and 0 < η 0 < 1. Theorem 2.5 relies on showing that the key assumption in Theorem [G] holds -finiteness of the sum of the L 2 -norms of the relevant conditional expectations. Showing that this assumptions holds reduces to showing that the sum of estimate (65) is bounded, and thus, we just need that f 0 h ∈ L . This again follows from (62).
The study of countable Markov partitions in the 1980's originated in particular from the work of Roy Adler [4]. His work motivated the use of countable Markov partitions as a tool for studying one-dimensional dynamics with critical points, and subsequently, Henon-like systems.
The first author keeps warmest memories of his visit in 1990 to IBM Thomas J Watson Research Center, when he worked within the wonderful group directed by Roy Adler.
The authors would like to thank Omri Sarig for useful discussions.