Statistical properties for compositions of standard maps with increasing coefficent
aa r X i v : . [ m a t h . D S ] O c t STATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPSWITH INCREASING COEFFICIENT
ALEX BLUMENTHAL
Abstract.
The Chirikov standard map family is a one-parameter family of volume-preservingmaps exhibiting hyperbolicity on a ‘large’ but noninvariant subset of phase space. Based on thispredominant hyperbolicity and numerical experiments, it is anticipated that the standard maphas positive metric entropy for many parameter values. However, rigorous analysis is notoriouslydifficult, and it remains an open question whether the standard map has positive metric entropy forany parameter value. Here we study a problem of intermediate difficulty: compositions of standardmaps with increasing parameter. When the coefficients increase to infinity at a sufficiently fastpolynomial rate, we obtain a Strong Law, Central Limit Theorem, and quantitative mixing estimatefor Holder observables. The methods used are not specific to the standard map and apply to a classof compositions of ‘prototypical’ 2D maps with hyperbolicity on ‘most’ of phase space. Introduction and statement of results
Let f : M → M be a smooth dynamical system. In many systems of interest, the dynamics of f does not tend to a stable or periodic equilibrium, as evidenced, e.g., when observables φ : M → R of such systems fluctuate indefinitely, i.e., φ ◦ f n ( x ) fluctuates as n → ∞ for a ‘large’ set of x ∈ M .In such cases, the asymptotic dynamics of the system is best described not by equilibria, but by a‘physical’ measure µ for f : an f -invariant probability measure µ on M is called physical if for apositive Lebesgue measure set of x ∈ M (the ‘basin’ of µ ) and any observable φ : M → R , we havethat lim n →∞ n n − X i =0 φ ◦ f i ( x ) = Z φ dµ . (1)Treating the sequence of observations { φ ◦ f i } i ≥ as a sequence of random variables, (1) aboveis a Strong Law of Large Numbers. Pursuing this interpretation, it is natural to ask whether finerstatistical properties hold, e.g.: • Central Limit Theorems pertaining to the convergence in distribution of √ n P n − i =0 ( φ ◦ f i ( X ) − m ), where X is distributed in M with some given law ν and m ∈ R is a cen-tering constant; and • Decay of Correlations, i.e., estimates on the decay of | R φ ◦ f n · ψ dµ − R φ dµ R ψ dµ | as n → ∞ for some class of observables φ, ψ on M .These properties are by now classical for maps f with uniform hyperbolicity, e.g., expanding,Anosov or Axiom A maps (see, e.g., [20]). Outside the ‘uniform’ setting, an extremely importanttool in the exploration of statistical properties of deterministic dynamical systems is nonuniformlyhyperbolic theory, also known as Pesin theory [6, 30]. Assuming some control on the (typicallynonuniform) rate of hyperbolicity, techniques have been developed for use in conjunction with Date : September 4, 2018.2010
Mathematics Subject Classification.
Primary: 37C60, 37A25, 37D25; Secondary: 60F05.
Key words and phrases. nonautonomous dynamics, nonuniform hyperbolicity, statistical properties of deterministicdynamics.This research was supported by NSF Grant DMS-1604805. nonuniform hyperbolicity to probe finer statistical properties of deterministic dynamical systems(e.g., the technique of countable Markov extensions, also known as Young towers [31]).
Difficulties and challenges.
Use of these tools requires establishing nonuniform hyperbolicity, which is notoriously difficultto verify even for maps which ‘appear’ to be hyperbolic on most (but not all) of phase space. In thevolume-preserving category, the difficulties involved are exemplified by the Chirikov standard mapfamily { F L } L> of volume-preserving maps on the torus T [12]. For large L , the map F L exhibitsstrong hyperbolicity (i.e., F L admits a continuous, invariant family of cones with strong expansion)on a large but noninvariant subset of phase space. A key difficulty is that typical orbits will entera set where cone invariance is violated (e.g. the vicinity of an elliptic fixed point for F L ), andthe previously expanding invariant cone is potentially ‘twisted’ towards the strongly contractingdirection, after which all the growth accumulated may be destroyed. Results in this paper.
In the interest of studying a problem of intermediate difficulty between the classical uniformlyhyperbolic settings and the presently intractable two-dimensional nonuniformly hyperbolic settingexemplified by the Standard Map, we propose to study compositions of standard maps with in-creasing coefficient. Cone twisting does occurs on a positive-volume subset of phase space at eachtimestep, and so we contend with many of the same problems described above for systems awayfrom the ‘uniform setting’. Indeed, our hypotheses do not preclude the existence of elliptic fixedpoints for our compositions. Important for our analysis, however, is the fact that increasing thecoefficient at each timestep both increases the strength of expansion and decreases the size of phasespace committing ‘cone twisting’– a crucial feature of this model is that a generic trajectory reachesthese ‘bad’ regions at most finitely many times when the increasing coefficients { L n } are inversesummable (see § fixed coefficient L , i.e., correlations estimates providingsharp bounds at all times n ≤ N L (in our results, N L grows as a fractional power of L ). Thisresult (formulated as Theorem D) is of independent interest: although it fails to be true asymptotic result, these estimates demonstrate that for large L , the Standard map F L is strongly mixing on arelatively long timescale. Related prior work.
The study of nonautonomous dynamical systems is still in its infancy, and many open questionsremain. That being said, the statistical properties explored in this paper are closest to those onmemory loss for nonautonomous compositions of hyperbolic maps [4, 5, 27] (see also [3]); Sinaibilliards systems with slowly moving scatterers [10, 28, 29]; and polynomial loss of memory forintermittent-type maps of the interval with a neutral fixed-point at the origin [1, 23]. We havebenefited especially from the techniques in [13], which studies statistical properties of sequentialpiecewise expanding compositions in one dimension.Pertaining to the Chirikov standard map, there is a large literature on this and related systems(e.g., Schroedinger cocycles) which we do not include here. See, e.g., the citations in [9] for a smallsampling of such results.
TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT 3
Random dynamical systems can be thought of as a version of nonautonomous dynamics withsome stationarity properties; see, e.g., [2, 19]. Lyapunov exponents of random perturbations of thestandard map with large coupling coefficient were studied in [9]. We also note [18], which establishedquenched (samplewise) statistical properties for a large class of SDE in both the volume-preservingand dissipative regimens.The analysis in this paper bears some qualitative similarities with that used in [8], which stud-ies Lyapunov exponents and statistical properties of random perturbations of dissipative two-dimensional maps with qualitatively similar features to the Henon map; these results apply aswell to the standard map. As it turns out, statistical properties of the corresponding Markov chaincan be deduced from finite-time mixing estimates for the dynamics, very much in keeping with thespirit of the analysis in the present paper (especially Theorem D).Lastly, we mention that the techniques in this paper may be useful in future studies of ‘bouncingball’ models of Fermi acceleration [14, 15, 17]. As it turns out, the static wall approximation ofbouncing ball models in a potential field gives rise to a Poincare return map bearing strong quali-tative similarities to the standard map (see [14] for a detailed derivation), and so it is conceivablethat the analysis in this paper may shed insight on open problems related to “escaping trajectories”for such models.
Acknowlegements.
The author thanks Dmitry Dolgopyat for suggesting this problem and for many helpful discussions.1.1.
Statement of results.
Definition of model.
Let M ∈ N , K , K > L >
0, which should bethought of as sufficiently large, and let { L n } be a nondecreasing sequence for which L ≤ L ≤ L ≤ · · · ≤ L n ≤ · · · for all n . In our results, we will assume that L n → ∞ at a sufficiently fastpolynomial rate in n .For each n ≥
1, let f n : T → R be a C function for which(H1) k f ′ n k C = k f ′ n k C + k f ′′ n k C ≤ K L n ,(H2) C n := { ˆ x ∈ T : f ′ n (ˆ x ) = 0 } is finite, with cardinality ≤ M , and(H3) For any n ≥ , x ∈ T we have | f ′ n ( x ) | ≥ L n K − d ( x, C n ).We will consider the nonautonomous composition of the maps F n : T → T defined by setting F n = (cid:18) f n ( x ) − y (mod 1) x (cid:19) . Above, (mod 1) refers to the projection R → T defined by x x − ⌊ x ⌋ , having abused notationsomewhat and parametrized T by [0 , f n ( x ) := L n sin(2 πx ) + 2 x ,in which case F n is (up to conjugation by a linear toral automorphism) the Standard map withcoefficient L n . These conditions are also satisfied for the family f n ( x ) := L n ψ ( x ) + a n where { a n } ⊂ [0 ,
1) is any subsequence and ψ : T → R is a map satisfying some C -generic conditions–details are left to the reader. The hypotheses (H1) – (H3) are similar to those for Theorem 1 in [9].For n ≥ m ≥
1, we write F nm = F n ◦ F n − ◦· · ·◦ F m , and write F n = F n . We adopt the conventions F n − n = Id, F = Id. Here for x ∈ R we define the floor function ⌊ x ⌋ = max { n ∈ Z : n ≤ x } . ALEX BLUMENTHAL
Results.
Our first result is a Strong Law of Large Numbers, which can be thought of as an ergodicity-type property for the nonautonomous compositions { F n } . Theorem A.
Let α ∈ (0 , . Assume that { L n } is nondecreasing, and that L ≥ L , where L = L ( K , K , M , α ) > is a constant. Let φ : T → R be α -Holder continuous with R φ d Leb T = 0 . (a) If N L − α α +4 N → , then N P Ni =0 φ ◦ F i → in L . (b) If N ǫ L − α α +4 N → for some arbitrary ǫ > , then N P Ni =0 φ ◦ F i → Lebesgue almost-everywhere.
Example 1.1.
Fix α ∈ (0 , , p >
0. Define L n = max { L , n p } for some p >
1. Then, TheoremA(a) holds when p > α − (6 α + 8) and (b) holds when p ≥ α − (12 α + 16). The results are optimalwhen α = 1 (i.e. φ is Lipschitz); here p ≥
14 suffices for (a) and p ≥
32 for (b).Next is a central limit theorem for Holder observables.
Theorem B.
Let α ∈ (0 , . Let { L n } be as in Theorem A, and additionally, assume lim N →∞ N L − α α +4 N = 0 . Let φ be an α -Holder continuous function on T for which R φ d Leb T = 0 . Let X be a uniformlydistributed T -valued random variable. Then, σ √ N P Ni =0 φ ◦ F i ( X ) converges in distribution to astandard Gaussian as N → ∞ with σ = Z φ ( x, y ) dxdy + 2 Z φ ( x, z ) φ ( z, y ) dxdydz , provided that σ > . Moreover, we have σ = 0 iff φ ( x, y ) = ψ ( x ) − ψ ( y ) for some continuous ψ : T → R . These conditions are satisfied for L n as in Example 1.2 when p > α − (3 α + 4). The asymptoticvariance σ appearing in Theorem B comes from an appropriate interpretation of the ‘singular’ limitof the maps F n as n → ∞ . The condition φ ( x, y ) = ψ ( x ) − ψ ( y ) has the connotation of a coboundarycondition for this singular limit. See the discussion in § §
5. In the setting of Example 1.2, Theorem B holdswhen p > α +4) α ; the result is optimal when α = 1, in which case p >
56 suffices.Finally, we present a decay of correlations estimate for the compositions { F n } . Theorem C.
Fix η ∈ (1 / , . Let { L n } be a nondecreasing sequence for which L ≥ L ′ , where L ′ = L ′ ( K , K , M , η ) is a constant, and assume that P n L − (1 − η ) n < ∞ for some fixed η ∈ (1 / , . Then, there is a constant C = C ( K , K , M , η ) for which the following holds.Let α ∈ (0 , and let ϕ, ψ be α -Holder continuous functions on T . Then, (cid:12)(cid:12)(cid:12)(cid:12) Z ψ ◦ F n · ϕ − Z ϕ Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k α k ϕ k α max (cid:26) L − η ⌊ n/ ⌋ , (cid:18) ∞ X i = ⌊ n/ ⌋ L − (1 − η ) i (cid:19) αα +2 (cid:27) for all n ≥ . Above, all integrals R are with respect to Leb T , and we have written[ ϕ ] α := sup p,q ∈ T | ϕ ( p ) − ϕ ( q ) | d T ( p, q ) α and k ϕ k α = k ϕ k C + [ ϕ ] α for α ∈ (0 ,
1] are Holder moduli and norms, respectively, and d T is the geodesic distance on T endowed with the flat geometry of R / Z . TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT 5
Example 1.2.
Fix η ∈ (1 / , L n = max { L ′ , n p } for some p >
4. In particular, P n L − (1 − η ) n < ∞ iff p > / (1 − η ). One obtains that the max {· · · } term on the right-handside is ≤ Const. k ψ k α k φ k α n max { p (1 − η ) , αα +2 (1 − p (1 − η )) } The exponent of n is optimized at η = αp +4 p − α αp +8 p at the value (4 − p ) α α +8 (valid since here p > / (1 − η )reduces to p >
4, which has been assumed), leading to the estimate ≤ Const. k ψ k α k φ k α n − α ( p − α +8 The result is strongest when α = 1, in which case decay of correlations is summable if p > Finite-time decay of correlations estimates for fixed-coefficient standard maps.
Our estimates in this paper can also be used to obtain the following finite time decay of correlationsestimate for Holder observables.
Theorem D.
Let α ∈ (0 , , and let L ≥ L ′′ , where L ′′ = L ′′ ( α ) > is a constant. Let φ, ψ be α -Holder-continuous functions on T . Then, (cid:12)(cid:12)(cid:12)(cid:12) Z φ ◦ F nL · ψ − Z φ Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k α k ψ k α · nL − α α +4 . for all n ≥ , where C = C ( α ) > is a constant independent of L, ψ, φ . For each fixed
L >
0, Theorem D provides a nontrivial upper bound on correlations for times n ≪ L α α +4 , and thus gives information on the mixing properties of the standard map in theso-called anti-integrable limit. Like before, the result is strongest at α = 1. Plan for the paper.
We collection preliminaries and basic hyperoblicity results in §
2, with an em-phasis on the geometry of iterates of curves roughly parallel to the strongly expanding direction(called horizontal curves ) for the dynamics.In § { F n } ; this verifies Theorem Dand also lets us provide a statistical description of the ‘singular’ limit of the maps F n as n → ∞ .In § § § §
7, is logically independent of § §
5; indeed, itshould not be surprising that the ‘finite-time mixing’ estimates in these sections do not yield thelong-time asymptotic correlation estimate in Theorem C. The proof of the latter requires a morecareful study of the ‘shape’ of iterates of small, sufficiently nice sets S ⊂ T . This is carried out in §
6, and the proof of Theorem C is completed in § Notation and conventions. We T parametrize as [0 ,
1) throughout the paper. The torus T carriesthe flat geometry of R / Z , and we identify all tangent spaces with the same copy of R . We write d T , d T for the geodesic metrics on T , T respectively.We repeatedly use big-O notation: a quantity β ∈ R is said to be O ( κ ) for some κ >
0, written β = O ( κ ), if there is a constant C >
0, depending only on the system parameters K , K , M , forwhich | α | ≤ Cκ . Similarly, the letter C is reserved for any positive constant depending only on theparameters K , K , M .We write Leb or Leb T for the Lebesgue measure on T , although unless otherwise stated, anyintegral R over T should be assumed to be with respect to Lebesgue. When γ ⊂ T is a C curve,we write Leb γ for the (unnormalized) induced Lebesgue measure on γ .Lastly: the parameter L > L is enlarged, it is done so in a way that dependsonly on the system parameters K , K , M and the auxiliary parameter η introduced below in § ALEX BLUMENTHAL
From this point forward, we will assume that { L n } , { f n } are as in (H1) – (H3), and that { L n } is a nondecreasing sequence. Predominant hyperbolicity
For all large n , the maps F n are predominantly hyperbolic , which is to say that the derivativemaps dF n exhibit strong expansion along roughly horizontal directions on an increasingly large(but non-invariant) proportion of phase space. Our purpose in this section is to make this ideaprecise and collect some preliminary results.In § L n → ∞ sufficiently fast,we show that the compositions { F n } possess nonzero (in fact, infinite) Lyapunov exponents atLebesgue-almost every point. On the other hand, the rate at which this hyperbolicity is expressedis nonuniform across phase space, and so in analogy with standard nonuniformly hyperbolic theoryin the stationary setting, we develop in § uniformity set to control this nonuniformity.In § § § Predominant hyperbolicity of maps F n . Let us begin by identifying subsets of phasespace where the maps F n exhibit uniformly strong hyperbolicity. For L > n ≥
1, define the critical strips S n,L = { ( x, y ) ∈ T : d ( x, C n ) ≤ K L − n L } , and note that by (H3), for ( x, y ) / ∈ S n,L we have | f ′ n ( x ) | ≥ L . For each n , outside S n,L we have that F n is strongly expanding in the horizontal direction: to wit, for any L sufficiently large ( L ≥ n ≥ , p / ∈ S n,L , the cone C h = { v = ( v x , v y ) ∈ R : | v y | ≤ | v x |} is preserved by ( dF n ) p , and all vectors in the cone are expanded by a factor ≥ L/ S n,L ) ≈ L/L n . Thus, for fixed L , the proportion of phase space T \ S n,L on which F n preserves and expands C h increases as n increases. When the sequence L n increases sufficiently rapidly, this implies an infinite Lyapunov exponent almost everywhere: Lemma 2.1.
Assume P ∞ n =1 L − n < ∞ . Then, lim n →∞ n log k dF np k = ∞ (2) for Leb -almost every p ∈ T .Proof. For each
L >
0, we have ∞ X n =1 Leb( F n − ) − S n,L = ∞ X n =1 Leb S n,L ≤ K M L ∞ X n =1 L − n < ∞ . The Borel Cantelli lemma thus applies to the sequence of sets { ( F n − ) − S n,L } n ≥ , and so the set S L = { p ∈ T : F n − p ∈ S n,L i.o. } has zero Lebesgue measure. Taking S = ∪ ∞ N =1 S N , it is nowsimple to check that (2) holds for all p ∈ T \ S . (cid:3) Let us emphasize, however, that the limit (2) is highly nonuniform in x , due to the fact thatthe critical strips S n,L have positive mass for all n ≥
1. We encode this nonuniformity in a wayanalogous to that of uniformity sets (alternatively called
Pesin sets ) for nonuniformly hyperbolicdynamics.
TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT 7
Construction of uniformity sets for the composition { F n } . For our purposes in this paper,it is expedient to ‘fatten’ the critical strips S n,L as follows. Let η ∈ (0 , n ≥ B n ( η ) = { ( x, y ) ∈ T : d ( x, C n ) ≤ K L − ηn } . For p = ( x, y ) / ∈ B n ( η ), we have | f ′ n ( x ) | ≥ L ηn . In particular, for such p , we have that ( dF n ) p preserves the cone C h and expands tangent vectors in C x by a factor ≥ L ηn . The parameter η dictates the proportion of expansion we recover in ( B n ( η )) c , hence the tradeoff: the larger η , hencethe more expansion we demand away from the bad set B n ( η ), but the larger the bad sets B n ( η )become. We note that η appears throughout the paper and is often fixed in advance; as such, forsimplicity we often write B n = B n ( η ).For p ∈ T , define τ ( p ) = 1 + max { m ≥ F m − p ∈ B m } = min { k ≥ F n − p / ∈ B n for all n ≥ k } ;in particular, for a given orbit { p n = F n p } , the derivative mapping ( dF n ) p n − is uniformly expandingalong the horizontal cone C h for all n ≥ τ ( p ). In this way, the setsΓ N = { τ ( p ) ≤ N } . can be thought of as uniformity sets for the composition { F n } n ≥ . Repeating the proof of Lemma2.1 yields the following. Lemma 2.2.
Fix η ∈ (0 , , and assume that P ∞ n =1 L − ηn < ∞ . Then, τ < ∞ almost surely, and ∪ N Γ N has full Lebesgue measure. Indeed, we have the estimateLeb { τ > N } ≤ ∞ X n = N Leb( F n − ) − B n = ∞ X n = N Leb B n = O (cid:18) ∞ X n = N L − ηn (cid:19) . Horizontal curves.
Curves roughly parallel to unstable directions, sometimes called u -curvesin the literature, are an effective and well-used tool for describing the mixing mechanism of hyper-bolic dynamical systems: the elongation of such curves under successive applications of hyperbolicdynamics leads to their proliferation through phase space, resulting in mixing. These ideas are stan-dard for (autonomous) smooth dynamical systems exhibiting hyperbolicity; see, e.g., [16, 24, 26].In the setting of this paper, horizontal curves play the role of u -curves. Although much of thematerial in this section is standard for iterates of a single map, we note that the maps F n in ourcompositions become more singular as n increases. So, it is important to ensure that the necessaryestimates (e.g. distortion control) do not worsen with n . For this reason, we re-prove below in § Definition 2.3. A horizontal curve is a connected C curve γ ⊂ T with the property that γ = { ( x, h γ ( x )) : y ∈ I γ } for some (open, proper) subarc I γ ⊂ T and some Lipschitz continuousfunction h γ : I γ → T with Lip h γ ≤ / { F nm , m ≤ n } when these curves areassumed to avoid the critical strips B n for each n . Lemma 2.5 is a distortion estimate betweentrajectories evolving on the same horizontal curve. Finally, Lemma 2.7 considers the time evolutionof sufficiently long horizontal curves which are allowed to meet bad sets. ALEX BLUMENTHAL
The following is description of the geometry of successive images of horizontal curves which donot meet the bad sets { B n } . Lemma 2.4 (Forward graph transform) . Fix η ∈ (0 , ; then, the following holds whenever L issufficiently large (depending on η ). Let N ≥ , and let γ ⊂ T be a C horizontal curve of the form γ = γ N = graph g N = { ( x, g N ( x )) : x ∈ I N } , where I N ⊂ R and g N : I N → R is a C function forwhich k g ′ N k C ≤ / and k g ′′ N k C ≤ .Let n > N , and assume that for all N ≤ k ≤ n − , we have that F k − N ( γ ) ∩ B k = ∅ . Then for each N ≤ k ≤ n , we have that γ k = F k − N ( γ ) is a horizontal curve of the form graph g k = { ( x, g k ( x )) : x ∈ I k } for an interval I k ⊂ T and a C function g k : I k → T for which (a) We have the bounds k g ′ k k C ≤ L − ηk − and k g ′′ k k C ≤ K L − η +1 k − ; and (b) for any p iN ∈ γ, i = 1 , , writing F k − N p iN = p ik , we have that k p k − p k k ≤ L ηk k p k +1 − p k +1 k . Proof.
The proof is a standard graph transform argument, which we recall here. It suffices todescribe the induction step, that is, the procedure for obtaining g k +1 from g k for N ≤ k ≤ n − F k : T → R × T by setting ˜ F k ( x, y ) = ( f k ( x ) − y, x ) (thatis, without the ‘ (mod 1)’ in the first coordinate). Projecting ˜ F k ( x, g k ( x )) to the first coordinateresults in a function ˜ f k : I k → R of the form ˜ f k ( x ) = f k ( x ) − g k ( x ).Since γ k ∩ B k = ∅ , we have | f ′ k | ≥ L ηk on I k , and so | ˜ f ′ k | ≥ L ηk − / > L > f k : I k → R is invertible on its image ˜ I k +1 . Defining I k +1 ⊂ T to be the projectionof ˜ I k +1 to T , we define g k +1 : I k +1 → T to be the (uniquely determined) mapping for which g k +1 (cid:0) ˜ f k ( x ) (mod 1) (cid:1) = x for all x ∈ I k . This completes the description of the induction step.The estimates in item (a) is now derived from the implicit derivatives g ′ k ( x ) = 1( f ′ k − − g ′ k − )( g k ( x )) and g ′′ k ( x ) = − ( f ′′ k − − g ′′ k − )( f ′ k − − g ′ k − ) ( g k ( x )) . The estimate in (b) follows from the bound | ( ˜ f k ) ′ | ≥ L ηk − / ≥ L ηk . All estimates require taking L sufficiently large depending on η . (cid:3) Next we obtain distortion estimates along forward iterates of horizontal leaves in the setting ofLemma 2.4.
Lemma 2.5.
Assume the setting of Lemma 2.4. Let p iN ∈ γ, i = 1 , , and write p in = F n − N p iN .Then (cid:12)(cid:12)(cid:12)(cid:12) log k ( dF n − N ) p N | T γ kk ( dF n − N ) p N | T γ k (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) L − ηN k p n − p n k (cid:1) . Remark 2.6.
The above bound is quite poor unless η ∈ (1 / , η ∈ (1 / , η , the stronger the decay of correlations estimate in Theorem C. It is likelythat lowering η is possible: one way to accommodate the distortion estimate in Lemma 2.5 is tofurther subdivide images of the curve γ into pieces of size ≪ L − ηn . TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT 9
Proof of Lemma 2.5.
Write p ik = F k − N ( p iN ) = ( x ik , y ik ). Let γ k = F k − N ( γ ) and g k : I k → T , I k ⊂ T be as in Lemma 2.4. Then, k ( dF k ) p ik | T γ k k = s g ′ k +1 ( x ik +1 )) g ′ k ( x ik )) | f ′ k ( x ik ) − g ′ k ( x ik ) | , and solog k ( dF n − N ) p N | T γ kk ( dF n − N ) p N | T γ k = 12 (cid:18) log 1 + ( g ′ N ( x N )) g ′ N ( x N )) + log 1 + ( g ′ n ( x n )) g ′ n ( x n )) (cid:19) + n − X k = N log f ′ k ( x k ) − g ′ k ( x k ) f ′ k ( x k ) − g ′ k ( x k ) . (3)For the first two terms, observe that for β , β ∈ [0 , ∞ ), we have the elementary bound | log(1 + β ) − log(1 + β ) | ≤ | β − β | , and so for k = N, n , we have (cid:12)(cid:12)(cid:12)(cid:12) log 1 + ( g ′ k ( x k )) g ′ k ( x k )) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ( g ′ k ( x k )) − ( g ′ k ( x k )) | ≤ | g ′ k ( x k ) − g ′ k ( x k ) | ≤ g ′′ k ) · | x k − x k | . Applying the expansion estimate along images of horizontal curves as in Lemma 2.4(a), | x k − x k | ≤ L − ηk | x k +1 − x k +1 | ≤ · · · ≤ L − ηk · · · L − ηn − | x n − x n | (4)and the estimate Lip( g ′′ k ) ≤ K L − ηk coming from Lemma 2.4, we obtain the following upperbound for the first two terms in (3):Lip( g ′′ N ) · | x N − x N | + Lip( g ′′ n ) · | x n − x n | ≤ K L − ηN (1 + L − ( n − N ) ηN ) | x n − x n | . Thus these terms are O ( L − ηN ).We now estimate the summation term in (3). With ˜ f k = f k − g k : I k → R as in the proof ofLemma 2.4, we have that | log ˜ f ′ k ( x k ) − log ˜ f ′ k ( x k ) | ≤ sup ζ ∈ I k | ˜ f ′′ k ( ζ ) | inf ζ ∈ I k | ˜ f ′ n ( ζ ) | · | x k − x k | ≤ K L − ηk | x k − x k | . Applying (4) and collecting, (cid:12)(cid:12)(cid:12)(cid:12) log ( ˜ f n − N ) ′ ( x N )( ˜ f n − N ) ′ ( x N ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ K (cid:18) n − X k = N L − ηk L ηk L ηk +1 · · · L ηn − (cid:19) | x n − x n |≤ K L − ηN (cid:18) n − X k = N L − ( n − − k ) ηN (cid:19) | x n − x n |≤ K L − ηN k p n − p n k when L is taken suitably large. This completes the estimate. (cid:3) The above results describe the dynamics of a horizontal curve γ which ‘avoids’ the bad sets { B n } for some amount of time. On the other hand, if a given horizontal curve is allowed to meet thebad sets along its trajectory, then we lose control over the geometry where these iterates meet badsets. Below we describe an algorithm for excising those parts of a curve which fall into the bad setand describe the geometry of the parts of γ with a ‘good’ trajectory.We say that a horizontal curve γ is fully crossing if I γ = (0 ,
1) (all notation here and below is asin Definition 2.3).
Lemma 2.7.
Fix η ∈ (1 / , . Let γ be a horizontal curve. Then, for any m ≥ , k ≥ m , thereis a set B km ( γ ) ⊆ γ and a partition (possibly empty) of ¯Γ km ( γ ) of F km ( γ \ B km ( γ )) into fully crossingcurves with the following properties. (a) For any ¯ γ ∈ ¯Γ km ( γ ) , we have k h ′ ¯ γ k C ≤ L − ηk . (b) We have the estimate
Leb γ ( B km ( γ )) = O (cid:18) k X i = m L − ηi (cid:19) . (c) For any ¯ γ ∈ ¯Γ km ( γ ) and any p, p ′ ∈ ( F km ) − ¯ γ , we have k ( dF km ) p | T γ kk ( dF km ) p ′ | T γ k = 1 + O ( L − ηm )When k = m , we write ¯Γ m ( γ ) = ¯Γ mm ( γ ) , B m ( γ ) = B mm ( γ ) for short.Observe that Lemma 2.7 is inherently limited in two ways: (i) it is a finite-time result : for agiven curve γ and fixed m ≥
1, we have B km ( γ ) = γ for all k sufficiently large; and (ii) if γ is tooshort, then we may even have γ = B m ( γ ). Proof of Lemma 2.7.
Below, ˜ F m : T → R × T is as defined in the proof of Lemma 2.4. To start,we define ¯Γ m ( γ ) , B m ( γ ) as follows.For each connected component γ i , ≤ i ≤ k , of γ \ B m , the image ˜ γ i = ˜ F m ( γ i ) is of theform graph ˜ h i where ˜ h i : ˜ I i → T for an interval ˜ I i ⊂ R of the form ( a i − r i , b i + s i ), where a i , b i ∈ Z , r i , s i ∈ [0 , a i = b i , then we set ¯Γ m ( γ ) = ∅ and B m ( γ ) = γ , checking that if this is indeed the case, thenLeb γ ( γ ) = O ( L − ηm ) follows.When, a i < b i , we define ¯Γ m ( γ ) to be the collection of curves of the form graph ˜ h i ( · + l ) (projectedto T ) for l = a i , · · · , b i −
1. We set B m ( γ ) = ( γ ∩ B m ) ∪ k [ i =1 ( ˜ F m ) − graph(˜ h i | ( a i − r i ,a i ) ∪ ( b i ,b i + s i ) ) . For each curve of the form ˆ γ = ( ˜ F m ) − (graph ˜ h i | ( a i − r i ,a i ) ), we haveLeb γ (ˆ γ ) = O ( L − ηm )since γ i ∩ B m = ∅ , and similarly for curves of the form ˆ γ = ( ˜ F m ) − (graph ˜ h i | ( b i ,b i + s i ) ). Combiningthis with the bound Leb γ ( γ ∩ B m ) = O ( L − ηm ), we concludeLeb γ ( B m ( γ )) = O ( L − ηm ) . Lastly, Item (c) holds for k = m by Lemma 2.5.Let us now describe the induction procedure for obtaining ¯Γ l +1 m ( γ ) , B l +1 m ( γ ) l < k , assuming that¯Γ lm ( γ ) and B lm ( γ ) have been defined and that item (c) holds for k = l . We define¯Γ l +1 m ( γ ) := [ ¯ γ ∈ ¯Γ lm ( γ ) ¯Γ l +1 (¯ γ ) , and B l +1 m ( γ ) = B lm ( γ ) ∪ ( F lm ) − [ ¯ γ ∈ ¯Γ lm ( γ ) B l +1 (¯ γ ) . Repeating the above steps until step l = k , we have that ¯Γ km ( γ ) is comprised of fully crossinghorizontal curves ¯ γ for which k h ′ ¯ γ k C ≤ L − ηk . Item (c) similarly follows by the distortion estimatein Lemma 2.5.It remains to estimate the size of B km ( γ ). We have for each m ≤ l < k thatLeb γ ( B l +1 m ( γ )) = Leb γ ( B lm ( γ )) + Leb γ ( F lm ) − [ ¯ γ ∈ ¯Γ lm ( γ ) B l +1 (¯ γ ) . TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT11
For each ¯ γ ∈ ¯Γ lm ( γ ), we have Leb ¯ γ B l +1 (¯ γ ) = O ( L − ηl +1 ), and soLeb γ ( F lm ) − [ ¯ γ ∈ ¯Γ lm ( γ ) B l +1 (¯ γ ) = X ¯ γ ∈ ¯Γ lm ( γ ) Leb ¯ γ ( F lm ) − ( B l +1 (¯ γ ))= (1 + O ( L − ηm )) X ¯ γ ∈ ¯Γ lm ( γ ) Leb γ (( F lm ) − ¯ γ ) · Leb ¯ γ ( B l +1 (¯ γ ))Leb ¯ γ (¯ γ )= O ( L − ηl +1 )having applied the distortion estimate in item (c) with k = l . This completes the estimate. (cid:3) Decay of correlations for curves.
The proliferation of horizontal curves throughout phasespace is a mixing mechanism for our system. The estimates below justify this in the followingsense: the Lebesgue mass along a given fully crossing horizontal curve spreads around throughoutphase space in such a way as to appoximate Lebesgue measure very closely for Holder-continuousobservables.
Proposition 2.8.
Let η ∈ (1 / , . Assume L ≥ ¯ L , where ¯ L = ¯ L ( M , K , K , η ) . Let γ be afully crossing horizontal curve, and let ψ : T → R be α -Holder continuous. For ≤ m ≤ n , wehave (cid:12)(cid:12)(cid:12)(cid:12) Z γ ψ ◦ F nm d Leb γ − Len( γ ) · Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k α (cid:18) L − α (1 − η ) / ( α +2) n + L − ηm + n − X k = m L − ηk (cid:19) Note that Proposition 2.8 does not stipulate any conditions on the summability of the tail of { L n } . Proof.
With ψ fixed, γ a fully crossing horizontal curve, let K ∈ N , to be specified shortly, and let ℓ = K − .Let I , · · · , I K denote the partition of [0 ,
1) into K intervals of length ℓ each. For 1 ≤ i, j ≤ K ,let R i,j = I i × I j . Note that with ψ i,j = inf { ψ ( p ) : p ∈ R i,j } , we have k ψ − X ≤ i,j ≤ K ψ i,j χ R i,j k L ∞ = O ( ℓ α k ψ k α ) . Thus ( ∗ ) := Z γ ψ ◦ F nm d Leb γ = O ( ℓ α k ψ k α ) + X ≤ i,j ≤ K ψ i,j Z γ χ R i,j ◦ F nm d Leb γ . Form ¯Γ n − m ( γ ) , B n − m ( γ ) as in Lemma 2.7, so that for each i, j -summand, we have Z γ χ R i,j ◦ F nm d Leb γ = O (cid:18) n − X k = m L − ηk (cid:19) + X ¯ γ ∈ ¯Γ n − m ( γ ) Z ¯ γ χ R i,j ◦ F n d Leb ¯ γ k dF n − m k ◦ ( F n − m ) − , so that( ∗ ) = k ψ k α · O (cid:18) ℓ α + n − X k = m L − ηk (cid:19) + X ≤ i,j ≤ K ψ i,j X ¯ γ ∈ ¯Γ n − m ( γ ) Z ¯ γ d Leb ¯ γ k dF n − m k ◦ ( F n − m ) − χ R i,j ◦ F n . By the distortion estimate in Lemma 2.7(c), the i, j, ¯ γ -summand equals(1 + O ( L − ηm )) · Leb γ (( F n − m ) − ¯ γ ) Z ¯ γ χ R i,j ◦ F n d Leb ¯ γ | {z } ( ∗∗ ) . To estimate ( ∗∗ ), observe that ¯ γ ∩ F − n R i,j = ¯ γ | j , where for a set S ⊂ T we write S | i = S ∩ ( I i × [0 , n (¯ γ | j ) and the set B n (¯ γ | j ). We obtain( ∗∗ ) = Z ¯ γ χ R i,j ◦ F n d Leb ¯ γ = O ( B n (¯ γ | j )) + X ¯ γ ′ ∈ ¯Γ n (¯ γ | j ) Z ¯ γ ′ d Leb ¯ γ ′ k dF n | T ¯ γ k ◦ F − n χ R i,j = O ( L − ηn ) + (1 + O ( L − ηn )) · X ¯ γ ′ ∈ ¯Γ n (¯ γ | j ) Leb ¯ γ ′ (¯ γ ′ | i ) · Leb ¯ γ ( F − n ¯ γ ′ )by the distortion estimate in Lemma 2.7(c). Since k h ′ ¯ γ ′ k C ≤ L − ηn for each ¯ γ ′ ∈ ¯Γ n (¯ γ | j ), we easilyestimate Leb ¯ γ ′ (¯ γ ′ | i ) = (1 + O ( L − ηn )) ℓ , so that( ∗∗ ) = O ( L − ηn ) + (1 + O ( L − ηn ))(1 + O ( L − ηn )) · ℓ · Leb ¯ γ (¯ γ | j \ B n (¯ γ | j ))= O ( L − ηn ) + (1 + O ( L − ηn )) · ℓ · Leb ¯ γ (¯ γ | j \ B n (¯ γ | j )) . Now, Leb ¯ γ ( B n (¯ γ | j )) = O ( L − ηn ), soLeb ¯ γ (¯ γ | j \ B n (¯ γ | j )) = Leb ¯ γ (¯ γ | j ) + O ( L − ηn ) = (1 + O ( L − ηn − )) ℓ + O ( L − ηn )= (cid:0) O ( L − ηn − + ℓ − L − ηn ) (cid:1) ℓ having used the estimate k h ′ ¯ γ k C ≤ L − ηn − . Consolidating our estimates,( ∗∗ ) = O ( L − ηn ) + (1 + O ( L − ηn )) · (cid:0) O ( L − ηn − + ℓ − L − ηn ) (cid:1) · ℓ = (1 + O ( L − ηn + L − ηn − + ℓ − L − ηn )) ℓ . This establishes the constraint ℓ − L − ηn ≪
1. Plugging the above estimate back into theexpression for ( ∗ ) and using this constraint gives( ∗ ) = k ψ k α · O (cid:18) ℓ α + n − X k = m L − ηk (cid:19) + (1 + O ( L − ηm + L − ηn − + ℓ − L − ηn )) Leb γ ( γ \ B n − m ( γ )) · X ≤ i,j ≤ K ψ i,j ℓ = k ψ k α · O (cid:18) ℓ α + n − X k = m L − ηk (cid:19) + (1 + O ( L − ηm + ℓ − L − ηn )) (cid:0) Len( γ ) + O (cid:18) n − X k = m L − ηk (cid:19)(cid:1) · Z ψ = k ψ k α · O (cid:18) ℓ α + n − X k = m L − ηk (cid:19) + (1 + O ( L − ηm + ℓ − L − ηn )) Len( γ ) · Z ψ = Len( γ ) · Z ψ + k ψ k α · O (cid:18) ℓ α + n − X k = m L − ηk + L − ηm + ℓ − L − ηn (cid:19) . On setting K = ℓ − = ⌈ L (1 − η ) / ( α +2) n ⌉ , the proof is complete. (cid:3) Singular limit of { F n } ; finite time mixing estimates Although the compositions { F n } are nonautonomous or ‘nonstationary’ by design, we argue inthis section that the individual maps F n do converge, in a sense to be made precise, to some station-ary process. This we formulate in a precise way in § F nm for m, n very TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT13 large, m ≤ n ; we state and prove these mixing estimates in § § F L , L > Singular limit of { F n } . As n increases, the maps F n ( x, y ) = ( f n ( x ) − y (mod 1) , x ) becomemore and more singular due to the fact that L n → ∞ ; in particular, lim n →∞ F n does not exist inany meaningful topology on diffeomorphisms of T . To motivate a meaningful convergence notion,let us consider the action in the x coordinate given by the map f n : T → T .Observe that for n extremely large, f n : T → T is predominantly an expanding map, and so inone time iterate the value of f n ( x ) , x ∈ T is increasingly sensitive to x ∈ T . Cast in a differentlight, f n is increasingly ‘randomizing’ on T , to the point where x and f n ( x ) are increasinglydecorrelated as n → ∞ . One might expect, then, that in the limit, f n ( x ) can be modeled by arandom variable independent of x . A step towards a precise formulation might be as follows: forsome class of continuous observables φ, ψ : T → R , we should expect thatlim n →∞ Z T φ ◦ f n ( x ) · ψ ( x ) = Z φ Z ψ . Morally speaking, we expect that when X is a random variable distributed in a ‘nice’ way on T ,we have that the joint law of the pair ( X, f n ( X )) converges, in some to-be-determined sense, to thejoint law of a pair ( X, Z ) for which Z is independent of X .Let us now return to the implications for the full maps F n : T → T and make things moreprecise. The above discussion motivates modeling F n for n large by a Markov chain { Z n = ( X n , Y n ) } defined as follows. Let β , β , · · · be IID random variables uniformly distributed on T . Given aninitial condition Z = ( X , Y ) ∈ T , we iteratively define Z n +1 = ( X n +1 , Y n +1 ) = ( β n +1 , X n ) . for n ≥
0. The form of this Markov chain agrees with the idea, argued above, that
X, f n ( X ) are“asymptotically independent” in the sense described above.Let P denote the transition operator associated with Z n , so that P (( x, y ) , A × B ) = Leb( A ) · δ x ( B )for Borel A, B ⊂ T , where δ x denotes the Dirac mass at x . Write P k for the k -th iterate of P . For φ : T → R , k ≥
1, we define P k φ : T → R by P k φ ( x, y ) = R P k (( x, y ) , d ¯ xd ¯ y ) φ (¯ x, ¯ y ). Proposition 3.1.
Fix k ≥ and let φ, ψ : T → R be continuous. Assume L m → ∞ as m → ∞ .Then, lim m →∞ Z ψ ◦ F m + k − m · φ = Z P k ψ · φ . That is, the maps F n converge to the Markov chain ( Z n ) n in the sense that the associated Koopmanoperators converge to the transition operator P for Holder observables in a way reminiscent of theweak operator topology. Proposition 3.1 is proved in § Remark 3.2.
The convergence described in Proposition 3.1 suggests that the asymptotic varianceof sums √ N P N − i =0 φ ◦ F i as in the Central Limit Theorem (Theorem B) should coincide with theasymptotic variance ˆ σ ( φ ) of √ N P N − i =0 φ ( Z i ) , Z ∼ Leb T . Developing the Green-Kubo formula for ˆ σ ( φ ), we obtain ˆ σ ( φ ) = E (cid:0) φ ( Z ) (cid:1) + 2 ∞ X l =1 E (cid:0) φ ( Z ) φ ( Z l ) (cid:1) = E (cid:0) φ ( Z ) (cid:1) + 2 E (cid:0) φ ( Z ) φ ( Z ) (cid:1) = Z φ + 2 Z φ ( x, y ) φ ( y, z ) dxdydz , where we have used the fact that Z k , Z are independent when k ≥
2. This is precisely the form of σ given in Theorem B. Here, E refers to the expectation where Z ∼ Leb T .This perspective also explains the ‘coboundary condition’ φ ( x, y ) = ψ ( x ) − ψ ( y ) for some bounded ψ : T → R . If φ has this form, then the sums in the CLT for this Markov chain telescope: φ ( Z ) + φ ( Z ) + · · · + φ ( Z n − ) = − ψ ( Y ) + ψ ( X n ), and so the asymptotic variance is zero. Let usnow check that this is also a necessary condition for the asymptotic variance ˆ σ ( φ ) to be zero. Lemma 3.3.
Let φ : T → R be a Holder continuous function with R φ dxdy = 0 . Then, ˆ σ ( φ ) = 0 iff φ ( x, y ) = ψ ( x ) − ψ ( y ) , where ψ : T → R is some Holder continuous function.Proof. We have the identityˆ σ ( φ ) = Z (cid:18) φ ( x, y ) + Z φ ( z, x ) dz − Z φ ( w, y ) dw (cid:19) dxdy , the verification of which is an elementary (albeit tedious) computation left to the reader. Now,ˆ σ ( φ ) = 0 implies φ ( x, y ) = ψ ( x ) − ψ ( y ) pointwise (since φ is continuous), where ψ ( x ) := − R φ ( z, x ) dx . (cid:3) Finite-time mixing estimates.
The limiting notion described in Proposition 3.1 is at itscore the statement that finite compositions F nm , m ≤ n are ‘mixing’ in the limit m, n → ∞ . Wewill, in fact, prove something much stronger: a concrete estimate on the correlation of ( x, y ) to F nm ( x, y ) for m, n large. Proposition 3.4.
Fix η ∈ (1 / , and α ∈ (0 , . Let L be sufficiently large, depending on α, η .Let m ≥ and let φ , φ : T → R be α -Holder continuous functions. Then, there exists a constant C > , depending only on K , K , M , such that the following hold. (a) We have (cid:12)(cid:12)(cid:12)(cid:12) Z φ ◦ F m · φ − Z φ ( x, z ) φ ( z, y ) dxdydz (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k α k φ k α L − min { η − , α (1 − η )2+ α } m (b) Let n > m . Then, (cid:12)(cid:12)(cid:12)(cid:12) Z φ ◦ F nm · φ − Z φ Z φ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k φ k α k φ k α (cid:18) L − min { α (1 − η ) / (2+ α ) , η − } m + n − X k = m +1 L − ηk (cid:19) . Observe that Proposition 3.1 follows easily from Proposition 3.4. Moreover, as we leave to the readerto check, the proof of Proposition 3.4 requires only that the sequence { L n } be nondecreasing , andso applies equally well in the case when L m = L m +1 = · · · = L n = L for some fixed L >
0. ThusTheorem D follows.Items (a) and (b) are proved separately in § § TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT15
Proof of Proposition 3.4(a).
Throughout § § I i , R i,j be as in the proofof Proposition 2.8, where ℓ = K − and K ∈ N will be specified at the end (twice, once for part (a)and again for part (b)).With α ∈ (0 ,
1] and φ , φ fixed, for l = 1 , φ li,j = inf R i,j φ l , so that k φ l − X i,j φ li,j χ R i,j k L ∞ = O ( k φ k α ℓ α ) . To begin, we estimate Z φ ◦ F m · φ = O ( k φ k α k φ k α ℓ α ) + X ≤ i,j,i ′ ,j ′ ≤ K φ i,j φ i ′ ,j ′ Z χ R i,j ◦ F m · χ R i ′ ,j ′ = O ( k φ k α k φ k α ℓ α ) + X ≤ i ,i ,i ≤ K φ i i φ i i Z χ R i i χ R i i ◦ F m , where in passing from the first line to the second we have used that F m ( R i,j ) ⊂ [0 , × I i .Fixing i , i , i , let y ∈ I i and set H = I i × { y } . Applying Lemma 2.7,( ∗ ) = Z H χ R i i ◦ F m d Leb H = O (cid:0) Leb H ( B m ( H )) (cid:1) + X ¯ γ ∈ ¯Γ m ( H ) Z ¯ γ d Leb ¯ γ k dF m | T H k ◦ F − m χ R i i d Leb ¯ γ = O ( L − ηm ) + X ¯ γ ∈ ¯Γ m ( H ) (1 + O ( L − ηm )) Leb H ( F − m ¯ γ ) · Z ¯ γ χ I i × [0 , d Leb ¯ γ , having used again that F m ( I i × [0 , ⊂ [0 , × I i to develop the integrand on the far right.Estimating Leb ¯ γ (¯ γ ∩ I i × [0 , O ( L − ηm )) ℓ (having used that k h ′ ¯ γ k C = O ( L − ηm )), we obtain( ∗ ) = O ( L − ηm ) + (1 + O ( L − ηm ))(1 + O ( L − ηm )) Leb H ( H \ B m ( H )) · ℓ = O ( L − ηm ) + (1 + O ( L − ηm ))(1 + O ( L − ηm ))( ℓ + O ( L − ηm )) · ℓ = ℓ (cid:0) O ( L − ηm + ℓ − L − ηm (cid:1) . Integrating over y ∈ I i , we concludeLeb( R i i ∩ F − m R i i ) = ℓ (1 + O ( ℓ − L − ηm + L − ηm )) . Summing now over 1 ≤ i , i , i ≤ K gives Z φ ◦ F m · φ = O ( k φ k α k φ k α ( ℓ α + ℓ − L − ηm + L − ηm )) + X ≤ i ,i ,i ≤ K φ i i φ i i ℓ = O ( k φ k α k φ k α ( ℓ α + ℓ − L − ηm + L − ηm )) + Z φ ( x, z ) φ ( z, y ) dxdydz . The proof is complete on setting K = ℓ − = (cid:24) L − η α m (cid:25) .3.2.2. Proof of Proposition 3.4(b).
All notation is as in the beginning of § ∗∗ ) = Z φ ◦ F nm · φ = O ( k φ k α k φ k α ℓ α ) + X ≤ i,j ≤ K φ i,j Z R i,j φ ◦ F nm . Fix 1 ≤ i, j ≤ K . For y ∈ I j , write H = H ( y ) = I i × { y } . Then Z R i,j φ ◦ F nm = Z y ∈ I j Z H ( y ) φ ◦ F nm d Leb H ( y ) dy . Developing the inner integral and applying Lemma 2.7, Z H ( y ) φ ◦ F nm d Leb H ( y ) = O ( k φ k Leb H ( y ) B m ( H ( y ))) + X ¯ γ ∈ ¯Γ m ( γ ( y )) Z ¯ γ d Leb ¯ γ k dF m | T H ( y ) k ◦ F − m φ ◦ F nm +1 = O ( k φ k C L − ηm ) + (1 + O ( L − ηm )) X ¯ γ ∈ ¯Γ m ( H ( y )) Leb H ( y ) ( F − m ¯ γ ) Z ¯ γ φ ◦ F nm +1 d Leb ¯ γ . The curves ¯ γ cross the full horizontal extent of T and so fall under the purview of Proposition2.8. Applying the estimate there, we obtain Z ¯ γ φ ◦ F nm +1 d Leb ¯ γ = Len(¯ γ ) Z φ + O ( k φ k α (cid:18) L − α (1 − η ) / (2+ α ) n + L − ηm +1 + n − X k = m +1 L − ηk (cid:19) )= Z φ + O ( k φ k α (cid:18) L − α (1 − η ) / (2+ α ) n + L − ηm +1 + n − X k = m +1 L − ηk (cid:19) ) . Summing over ¯ γ we obtain that R H ( y ) φ ◦ F nm d Leb H ( y ) equals O ( k φ k C L − ηm ) + (1 + O ( L − ηm )) X ¯ γ ∈ ¯Γ m ( H ( y )) Leb H ( y ) ( F − m ¯ γ ) · (cid:18) Z φ + O ( k φ k α (cid:18) L − α (1 − η ) / (2+ α ) n + L − ηm +1 + n − X k = m +1 L − ηk (cid:19) ) (cid:19) = O ( k φ k α (cid:18) L − ηm + ℓ ( L − α (1 − η ) / (2+ α ) n + L − ηm +1 + n − X k = m +1 L − ηk ) (cid:19) )+(1 + O ( L − ηm )) Leb H ( y ) ( H ( y ) \ B m ( H ( y ))) Z φ = O ( k φ k α (cid:18) L − ηm + ℓ ( L − α (1 − η ) / (2+ α ) n + L − ηm +1 + n − X k = m +1 L − ηk ) (cid:19) )+(1 + O ( L − ηm ))(1 + O ( ℓ − L − ηm )) · ℓ Z φ = ℓ · (cid:26) O ( k φ k α (cid:18) ℓ − L − ηm + L − α (1 − η ) / (2+ α ) n + L − ηm + n − X k = m +1 L − ηk (cid:19) ) + Z φ (cid:27) . Integrating over y ∈ I j yields the same estimate for R χ R i,j φ ◦ F nm with an additional factor of ℓ . Summing over 1 ≤ i, j ≤ K , we have that R φ ◦ F nm · φ equals O ( k φ k α k φ k α (cid:18) ℓ α + ℓ − L − ηm + L − α (1 − η ) / (2+ α ) n + L − ηm + n − X k = m +1 L − ηk (cid:19) ) + K X i,j =1 ℓ φ i,j Z φ = O ( k φ k α k φ k α (cid:18) ℓ α + ℓ − L − ηm + L − α (1 − η ) / (2+ α ) n + L − ηm + n − X k = m +1 L − ηk (cid:19) ) + Z φ Z φ . The proof is complete on setting K = ⌈ L (1 − η ) / (1+ α ) m ⌉ . TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT17 Law of Large Numbers
We continue our study of the statistical properties of the composition { F n } by proving TheoremA, a pair of formulations of a ‘law of large numbers’ for time-averages of observables. In this section, α ∈ (0 , is fixed, as are a sequence of α -Holder continuous observables φ i : T → R , i ≥ with R φ i = 0 for all i and sup i ≥ k φ i k α ≤ C for a constant C > . For 0 ≤ M ≤ N , we define ˆ S M,N = φ M ◦ F M + · · · + φ N ◦ F N and set ˆ S N = ˆ S ,N . Noting the simple estimate | ˆ S N − ˆ S M,N | = (cid:12)(cid:12)(cid:12)(cid:12) M − X i =0 φ i ◦ F i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C M holds pointwise on T , it follows that to prove a strong law for ˆ S N , it suffices to prove a strong lawfor ˆ S M,N where M = M ( N ) = ⌊√ N ⌋ . Similarly, a weak law for ˆ S N follows from a weak law forˆ S M,N . More precisely, to prove Theorem A it suffices to prove the following.
Proposition 4.1.
For N ≥ let M = M ( N ) = ⌊√ N ⌋ . (a) If N L − α α +4 N → , then N − M ˆ S M,N converges in L to . (b) If N ǫ L − α α +4 ⌊√ N ⌋ → as N → ∞ for some ǫ > , then N − M ˆ S M,N converges almost surely to .Proof of Proposition 4.1. To start, we expand Z ˆ S M,N = N X n = M Z φ n ◦ F nM + 2 X M ≤ m 0, as we have in the hypotheses of item (a). For (b), our estimates imply that the sequence { N − R ˆ S M,N } N ≥ is summable whenever N ǫ L − α α +4 ⌊√ N ⌋ → ǫ > (cid:3) Central limit theorem Here we carry out the proof of of the central limit theorem in Theorem B. A standard technique,attributed to Gordin, for proving the central limit theorem for a deterministic dynamical systemis to look for reverse Martingale difference approximations for sums of observables, and then touse probability theory tools for proving the Central Limit Theorem for sums of reverse Martingaledifferences (see, e.g., [21] for an exposition).We pursue a slightly different method: we construct here an array of forward Martingale differ-ence approximations. The corresponding forward filtrations are comprised (mostly) of fully-crossinghorizontal curves. The filtration is constructed in § § § § Throughout this section, α ∈ (0 , is fixed, and φ : T → R is assumed to be an α -Holder continuousobservable with R φ = 0 . The value η ∈ (1 / , is assumed fixed; as we did in the previous section,in § η depending on α . Notation: We write E below for the expectation with respect to Lebesgue measure on T . When G is a sub-sigma-algebra of the Borel sigma algebra, we write E ( ·|G ) for the conditional expectationwith respect to G .5.1. Preliminaries for CLT: Construction of a martingale approximation. Construction of the increasing filtrations { ˆ G M,k , k ≥ M } . We will produce an increasingfiltration of (most of) T by horizontal curves with a small and controlled exceptional set. Below, M ∈ N should be thought of as large.First, we will construct a sequence of partitions ζ M,M , ζ M,M +1 , · · · , ζ M,k , · · · of T with thefollowing properties for each M ≤ k ≤ N :(A) The partition ζ M,k is “mostly” comprised of fully crossing horizontal curves; and(B) ζ M,k ≤ F − k ζ M,k +12 .Once the ζ M,k are constructed, we define G M,k to be the sigma algebra of measurable unions ofelements in ζ M,k , and finally, ˆ G M,k = ( F kM ) − G M,k +1 , so that { ˆ G M,k } k ≥ M is an increasing filtration on T . This is the filtration we will use in the sequelto construct our forward Martingale difference approximation. Construction of { ζ M,k , k ≥ M } satisfying (A), (B). Set ζ M,M to be the partition of T \ { x = 0 } into horizontal line segments. Applying Lemma 2.7, for each ζ ∈ ζ M,M form B M ( ζ ) and ¯Γ M ( ζ ),writing G M,M +1 = [ ζ ∈ ζ M,M ¯ ζ ∈ ¯Γ M ( ζ ) ¯ ζ , B M,M +1 = [ ζ ∈ ζ M,M F M ( B M ( ζ )) . Defining the partition H M,M +1 = { G M,M +1 , B M,M +1 } , we now define the partition ζ M,M +1 ≥H M,M +1 as follows: ζ M,M +1 | G M,M +1 = { ¯ ζ : ¯ ζ ∈ ¯Γ M ( ζ ) , ζ ∈ ζ M,M } ,ζ M,M +1 | B M,M +1 = { F M ( ζ ) ∩ B M,M +1 : ζ ∈ ζ M,M } . Here “ ≤ ” refers to the partial order on partitions: two partitions ζ, ζ ′ satisfy ζ ≤ ζ ′ if any atom of ζ is a unionof ζ ′ atoms. TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT19 Iterating, assume ζ M,k has been formed, where k ≥ M + 2, along with the partition H M,k = { G M,k , B M,k } for which ζ M,k ≥ H M,k . For each ζ ∈ ζ M,k | G M,k form ¯Γ k ( ζ ) and define G M,k +1 = [ ζ ∈ ζ M,k ¯ ζ ∈ ¯Γ k ( ζ ) ¯ ζ , B M,k +1 = [ ζ ∈ ζ M,k F k ( B k ( ζ )) , and define ζ M,k +1 by ζ M,k +1 | G M,k +1 = { ¯ ζ : ¯ ζ ∈ ¯Γ k ( ζ ) , ζ ∈ ζ M,k | G M,k } ,ζ M,k +1 | B M,k +1 = { F k ( ζ ) ∩ B M,k +1 : ζ ∈ ζ M,k } . Below, we formulate and verify properties (A) and (B) above for the sequence ζ M,k , k ≥ M con-structed above. Lemma 5.1. The partitions { ζ M,k } k ≥ M , H M,k = { G M,k , B M,k } are measurable, and have the fol-lowing properties for each k ≥ M . (a) Every atom ζ ∈ ζ M,k | G M,k is a fully crossing horizontal curve for which k h ′ ζ k C ≤ L − ηk − . (b) We have ζ M,k ≤ F − k ζ M,k +1 . (c) We have the estimate: Leb( B M,k ) = O (cid:18) k − X i = M L − ηi (cid:19) . Proof. Measurability is not hard to check. Items (a) and (b) follow from the construction. Forthe estimate in item (c), observe that for each k ≥ M , ζ ∈ ζ M,k | G M,k , we have Leb ζ ( B k ( ζ )) = O ( L − ηk ), hence (Leb T ) ζ ( B k ( ζ )) ≤ (1 + O ( L − ηk − )) · O ( L − ηk ) = O ( L − ηk ), where here (Leb T ) ζ isthe disintegration measure of Leb T | G M,k with respect to ζ ∈ ζ M,k | G M,k . We concludeLeb( G M,k +1 ) = (1 + O ( L − ηk )) Leb( G M,k ) , hence Leb( G M,m ) = m − Y k = M (1 + O ( L − ηk )) ≥ O (cid:18) m − X k = M L − ηk (cid:19) . (cid:3) The choice of ˆ G M,k is made so that F k − M ˆ G M,k = F − k G M,k +1 is a very ‘fine’ sigma-algebra. Beforeproceeding, we record the following estimate. Lemma 5.2. Let φ be α -Holder continuous, k ≥ M . Then | φ − E ( φ | F − k G M,k +1 ) | = O ( k φ k α L − ηαk ) . on F − k G M,k +1 .Proof. Let ζ ∈ G M,k +1 | G M,k +1 . Then F − k ζ is, by our construction, a subsegment of a fully-crossingcurve ζ ′ ∈ ζ M,k | G M,k with diameter O ( L − ηk ). So, for any points p, p ′ ∈ F − k ζ , we have | φ ( p ) − φ ( p ′ ) | = O ( k φ k α L − ηαk ). (cid:3) Approximation by sum of martingale differences. For a bounded observable φ : T → R ,convergence in distribution of √ N S N ( X ) , X ∼ Leb T , where S N = N X n =1 φ ◦ F n − , is equivalent to convergence in distribution of √ N S M,N ( X ) , X ∼ Leb T , where S M,N = N X n = M φ ◦ F n − M . and M = M ( N ) is a sequence satisfying M ( N ) ≪ √ N . Here, “ X ∼ Leb T ” means that X is a T -valued random variable with law Leb T .Thus, for Theorem B, it suffices to prove convergence in distribution of √ N S M,N ( X ); for this,we approximate S M,N by a sum of Martingale differences with respect to the increasing filtrationsˆ G M,k , k ≥ M . Proposition 5.3. Let M ≤ N . Define ˜ S M,N = N X n = M E ( φ | ( F n ) − G M,n +1 ) ◦ F n − M = N X n = M E ( φ ◦ F n − M | ˆ G M,n ) . (a) The sum ˜ S M,N admits the representation ˜ S M,N = P Nn = M U M,N,n , where U M,N,n = N − X m = n − (cid:18) E ( φ ◦ F mM | ˆ G M,n ) − E ( φ ◦ F mM | ˆ G M,n − ) (cid:19) . The sequence { U M,N,n , M ≤ n ≤ N } is a forward Martingale difference adapted to ( ˆ G M,n , M ≤ n ≤ N ) . Precisely, E ( U M,N,n | ˆ G M,n ) = U M,N,n and E ( U M,N,n | ˆ G M,n − ) = 0 . (b) We have | S M,N − ˜ S M,N | = O (cid:18) ( N − M ) k φ k α N X m = M L − ηαm (cid:19) on G M,N . Above, we use the convention that ˆ G M,M − = {∅ , T } is the trivial sigma-algebra on T . Fornotational simplicity, when M, N are fixed we write U n = U M,N,n . Proof. Item (b) is a simple consequence of Lemma 5.2. For item (a), the relation ˜ S M,N = P M ≤ n ≤ N U M,N,n can be verified by a direct computation.Alternatively, following the analogue of the derivation of a reverse Martingale difference ap-proximation given in [13] for forward martingale differences, one can look for a Martingale differ-ence U n = E ( φ ◦ F nM | ˆ G M,n ) + h n − h n +1 , where ( h n ) M ≤ n ≤ N +1 is some sequence of “cobound-ary” functions to be determined. Making the ansatz h N +1 = 0 and ‘solving’ the conditions E ( U n | ˆ G M,n ) = U n , E ( U n | ˆ G M,n − ) = 0 for each n , we deduce formally that h n = − N − X m = n − E (cid:0) φ ◦ F mM (cid:12)(cid:12) ˆ G M,n − (cid:1) . Plugging this formula into the relation U n = E ( φ ◦ F nM | ˆ G M,n ) + h n − h n +1 yields the form of U n given above. The choice ˆ G M,M − = {∅ , T } ensures that h M = 0, hence ˜ S M,N = P M ≤ n ≤ N U n + h M − h N +1 = P M ≤ n ≤ N U n holds. (cid:3) TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT21 Deducing Theorem B from the martingale approximation. We will deduce Theorem B fromthe following. Proposition 5.4. Assume N L − α α +4 N → as N → ∞ . For N > let M = M ( N ) = ⌊ √ N ⌋ . Then qP Nn = M E U M,N,n N X n = M U M,N,n ( X ) , X ∼ Leb T converges weakly to a standard Gaussian as N → ∞ . Proposition 5.4 is proved in the next section. Let us first complete the proof of Theorem B. Throughout, M = ⌊ √ N ⌋ . For the remainder of § 5, we specialize to the value η = α +23 α +4 , not-ing that this value maximizes the function η min { η − , α (1 − η )( α + 2) } . In particular, N L − min { η − ,α (1 − η ) / ( α +2) } N → as N → ∞ under the conditions of Proposition 5.4. As we noted in at the beginning of § √ N S M,N , since here M ≈ N / ≪ √ N . Thus, to prove Theorem B, it suffices to check that(I) k S M,N − ˜ S M,N k L → N → ∞ , and(II) N P Nn = M E U M,N,n → σ as N → ∞ , where σ is as in Theorem B.For (I), we estimate k S M,N − ˜ S M,N k L as follows: k S M,N − ˜ S M,N k L ≤ C ( N − M ) k φ k Leb( B M,N ) + C ( N − M ) k φ k α N X m = M L − ηαm ≤ C ( N − M ) k φ k α N X m = M L − min { αη, − η } m applying first Proposition 5.3(b) and then Lemma 5.1. The above converges to 0 as N → ∞ by thehypotheses of Proposition 5.4.For (II), we observe N X n = M E U M,N,n = E (cid:18) N X n = M U M,N,n (cid:19) = Z T ˜ S M,N d Leb T = Z T S M,N d Leb T + O ( k S M,N − ˜ S M,N k L ) . From (I), it follows that lim N →∞ (cid:0) k ˜ S M,N k L − k S M,N k L (cid:1) = 0. It remains to compute k S M,N k L ,which we do below. Lemma 5.5. Assume the setting of Proposition 5.4. With M = M ( N ) = ⌊ √ N ⌋ , we have lim N →∞ N Z S M,N d Leb = σ = Z φ + 2 Z φ ( x, z ) φ ( z, y ) dxdydz , Proof. We have Z S M,N = ( N − M + 1) Z φ + 2 X M ≤ m Proof of Proposition 5.4. We use the following criterion for the CLT for arrays of martingaledifferences. Theorem 5.6 (McLeish) . Let (Ω , F , P ) be a probability space. Let { k n } n ≥ , be an increasingsequence of whole numbers tending to infinity, and for each n ≥ , let F ,n ⊂ F ,n ⊂ · · · ⊂ F k n ,n ⊂F be an increasing sequence of sub- σ algebras. For each such n, i , let X i,n be a random variable,measurable with respect to F i,n , for which E ( X i,n |F i − ,n ) = 0 , and write Z n = P ≤ i ≤ k n X i,n .Assume (a) max i ≤ k n | X i,n | is uniformly bounded, in n , in the L norm, (b) max i ≤ k n | X i,n | → in probability as n → ∞ , and (c) P i X i,n → in probability as n → ∞ .Then, Z n converges weakly to a standard Gaussian. We apply this to the array1 qP Nm = M ( N ) E U M ( N ) ,N,m U M ( N ) ,N,n ( X ) , M ( N ) ≤ n ≤ N , X ∼ Leb T , where as before M ( N ) = ⌊ √ N ⌋ .A preliminary asymptotic estimate for U n is given in § § An asymptotic estimate for U n . The following approximation is extremely useful in the com-ing arguments. Lemma 5.7. Set ˆ U n = U n ◦ ( F n − M ) − . Then ˆ U n = φ − ψ + ψ ◦ F n +1 + O ( N k φ k α L − min { α (1 − η ) / (2+ α ) , η − } M ) with uniform constants on F − n G M,n +1 , independently of n , where ψ ( y ) = R φ (¯ x, y ) d ¯ x .Proof. We have ˆ U n = E ( φ | F − n G M,n +1 ) − E ( φ |G M,n ) + E ( φ |G M,n +1 ) ◦ F n + N − X m = n +1 E ( φ ◦ F mn +1 |G M,n +1 ) ◦ F n + N − X m = n E ( φ ◦ F mn |G M,n )(5)As we will show, the terms in the top line approximate to φ − ψ + ψ ◦ F n +1 , while the terms in thesecond line are small. TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT23 For the first term in (5), we have from Lemma 5.2 that | E ( φ | F − n G M,n +1 ) − φ | = O ( k φ k α L − αηn )on F − n G M,n +1 .For the second term in (5), we have that E ( φ |G M,n ) = 1Len( γ ) Z γ φ d Leb γ on G M,n , where γ is a fully crossing horizontal curve with k h ′ γ k C ≤ L − ηn − . Let now p ∈ γ , p = ( x , y ). Noting | φ ( x, h γ ( x )) − φ ( x, y ) | ≤ k φ k α | h γ ( x ) − h γ ( x ) | α ≤ C k φ k α L − αηn − , we have1Len( γ ) Z γ φd Leb γ = (1 + O ( k φ k α L − ηn − )) Z φ ( x, h γ ( x )) dx = (1 + O ( k φ k α L − αηn − )) Z φ ( x, y ) dx ;we therefore conclude | E ( φ |G M,n ) − ψ | ≤ C k φ k α L − αηn − on G M,n . Similarly, for the third term in (5), we obtain the bound | E ( φ |G M,n +1 ) ◦ F n − ψ ◦ F n | ≤ C k φ k α L − αηn on F − n G M,n +1 .For the fourth term in (5), we estimate from Proposition 2.8 that on G M,n , E ( φ ◦ F mn |G M,n ) = 1Len( γ ) Z γ φ ◦ F mn d Leb γ = O ( k φ k α · (cid:18) L − α (1 − η ) / (2+ α ) m + L − ηn + m − X k = n L − ηk (cid:19) )= O ( k φ k α ( N − M ) L − min { α (1 − η ) / (2+ α ) , η − } M )for some γ ∈ ζ M,n . Estimating similarly the fifth term in (5), we deduce that on F − n G M,n +1 thecontribution of the fourth and fifth terms combined is O ( k φ k α ( N − M ) L − min { α (1 − η ) / (2+ α ) , η − } M ) . (cid:3) Verifying properties (a) – (c) in Theorem 5.6.Properties (a) & (b). By Lemma 5.7, we have that on ( F N − M ) − G M,N , | U n | = O ( k φ k C + k φ k α N L − min { α (1 − η ) / (2+ α ) , η − } M ) = O ( k φ k α N L − min { α (1 − η ) / (2+ α ) , η − } M ) , which is uniformly bounded in n, N . Property (b) is now immediate, since Leb( G M,N ) → N → ∞ .For property (a), off ( F N − M ) − G M,N we have | U n | ≤ CN k φ k α , so, k max M ≤ n ≤ N | U M,N,n |k L ≤ C k φ k α · N q Leb( G cM,N ) + C k φ k α . Property (a) follows from the estimate Leb( G cM,N ) = O ( P N − M L − ηk ) = O (( N − M ) L − ηM ) inLemma 5.1. (cid:3) Below is a formulation of property (c). Proposition 5.8 (Strong law for { U n } ) . We have lim N →∞ P Nn = M U M,N,n E P Nn = M U M,N,n = 1 in probability. Proof. We prove the stronger property of convergence in L . To start, we evaluate Z (cid:18) N X M U n − N X M E ( U n ) (cid:19) d Leb = X M ≤ m,n ≤ N Z ( U n − E ( U n ))( U m − E ( U m )) d Leb= N X n = M (cid:0) E ( U n ) − E ( U n ) (cid:1) + 2 X M ≤ m Summing over the ≈ N terms and noting that (cid:0) P NM E ( U n ) (cid:1) ≈ σ N for N large, we obtain1 k φ k α (cid:13)(cid:13)(cid:13)(cid:13) P NM U n P NM E U n − (cid:13)(cid:13)(cid:13)(cid:13) L = O (cid:0) (1 + N L − ηM )( N L − ηM + N L − cM ) + N − + N L − ηM (cid:1) . The proof goes through if all terms on the RHS go to 0 as N → ∞ . For this, it suffices that N L − cM → N → ∞ : to see this, observe that N L − ηM ≤ N L − cM holds for any η ∈ (1 / , , α ∈ (0 , N L − cM → (cid:3) Hyperbolicity and the shape of successive iterates of a set We close this paper with the proof of Theorem C, given in § § § { F n } . In this section, we flesh out this picture by showing the following: given a set S ⊂ T witha suitably nice boundary and n large enough, the n -th image F n ( S ) is ‘mostly’ foliated by disjointfully-crossing horizontal curves.The plan is as follows. In § n a foliation of S n = F n − S by horizontalcurves. It is shown in § n sufficiently large, a large proportion of the curves in the foliationof S n are ‘sufficiently long’, in the sense that in one timestep such curves become fully crossing. In § S n , the disintegration densitieson the leaves of our horizontal foliation are controlled. These results are synthesized in Proposition6.11 in § § Construction of foliations by horizontal curves. Let S ⊂ T be an open subset, andwrite ν S for normalized Lebesgue measure on S . Our aim is to build a foliation of the n -th image F n ( S ) by horizontal curves with the property that for n sufficiently large, ‘most’ of the foliatingcurves are sufficiently long.6.1.1. Standing assumptions for § The parameter η ∈ (1 / , 1) is fixed. The open set S ⊂ T issuch that the topological boundary ∂S = ¯ S \ S is the finite union of smooth curves, and moreover,is assumed to have the following property: for any l > ν S { p ∈ S : d ( p, ∂S ) ≤ l } ≤ C S l , (6)where C S > l . Let us write S = S and F n − S = S n for n ≥ ∂S n = F n − ∂S since each F n is a diffeomorphism.For n ≥ 1, we write B n for the partition of T into the connected components of B n and B cn ,noting that each is a partition of T into vertical cylinders (sets of the form I × T for a properconnected subinterval I ⊂ T . We also abuse notation somewhat and write ∂ B n for the union of theboundaries of each atom of B n ; that is, ∂ B n is the union of circles of the form { ˆ x n ± K L − ηn }× T as ˆ x n varies over C n .Define the sequence of partitions {P n } n ≥ of T as follows: P = B ∨ { S , S c } , and for n ≥ P n = B n ∨ F n − ( P n − ) . Above, ∨ refers to the join of partitions. Hereafter for q ∈ T , we write P n ( q ) for the atom of P n containing q . Again we abuse notation somewhat and write ∂ P n for the union over the collectionof boundaries of each atom comprising P n . Additional notation: For q = ( x, y ) ∈ T , let us write H q = T × { y } for the horizontal circlecontaining q . When P is a partition of T and p ∈ T , we write P ( p ) for the atom of P containing p . We write “ ≤ ” for the partial order on partitions: for partitions P , Q , we write P ≤ Q if eachatom in P is a union of Q -atoms.6.1.2. Algorithm for foliating S n by horizontal curves. We now define, for each n ≥ 1, a foliation(partition) ˆ γ n of S n by horizontal curves.For n = 1, we define ˆ γ to be the partition of S consisting of atoms of the formˆ γ ( p ) = H p ∩ P ( p )for p ∈ S . Clearly ˆ γ is a measurable partition of S , and ˆ γ ≤ P | S (here ≤ indicates the partialorder on partitions in terms of refinement, and P | S denotes the restriction of P to S ). Induc-tively, assume that ˆ γ , · · · , ˆ γ n have been constructed, and that ˆ γ n ≥ P n | S n . To define ˆ γ n +1 ( p n +1 )for p n +1 ∈ S n +1 , we distinguish two cases. Below we write p n = F − n ( p n +1 ). Case 1: p n / ∈ B n . By construction, ˆ γ n ( p n ) ∩ B n = ∅ , and so F n (ˆ γ n ( p n )) is a horizontal curve(Lemma 2.4). In preparation for the next iterate, we cut this image curve by P n +1 ; that is,ˆ γ n +1 ( p n +1 ) = F n (ˆ γ n ( p n )) ∩ P n +1 ( p n +1 ) . Equivalently, ˆ γ n +1 | F n ( B cn ∩ S n ) = F n (ˆ γ n ∩ B cn ) ∨ P n +1 | F n ( B cn ∩ S n ) Case 2: p n ∈ B n . In this case ˆ γ n ( p n ) ⊂ B n and so we lose our control on the image curve F n (ˆ γ n ( p n )). The procedure here is to re-partition the entire image of B n by horizontal line segmentscut by P n +1 , in preparation for the next iterate. Precisely, we defineˆ γ n +1 ( p n +1 ) = H p n +1 ∩ P n +1 ( p n +1 ) . Equivalently, ˆ γ n +1 | F n ( B n ∩ S n ) is the join of P n +1 | F n ( B n ∩ S n ) with the partition of F n ( B n ) into hori-zontal circles (sets of the form T × { y } ⊂ T for y ∈ T ).This induction procedure bootstraps because ˆ γ n +1 is a partition of S n +1 into horizontal curvesfor which ˆ γ n +1 ≥ P n +1 | S n +1 . All partitions mentioned are measurable [25], and so we have thefollowing. Lemma 6.1. For each n ≥ , the partition ˆ γ n of S n as above is defined and is a measurablepartition of S n into connected, smooth horizontal curves for which ˆ γ n ≥ P n | S n . Estimating time to curve length growth. As indicated in the procedure laid out above,the curves of ˆ γ n +1 coming from ˆ γ n | S n ∩ B cn have been elongated by the strong expansion of F n alonghorizontal directions. However, this elongation of curves competes with the ‘cutting’ of curvesnear bad sets (case 1) and the occasional ‘repartitioning’ of the images of the bad sets S n ∩ B n byhorizontal line segments (case 2). Our aim now is to show that for large n , the expansion wins out,and ‘most’ of the curves comprising the foliation ˆ γ n are of sufficiently long horizontal extent.6.2.1. Preparations. For a connected C curve γ ⊂ T and a point q = ( x, y ) ∈ γ , we defineRad q ( γ ) = d γ ( q, ∂γ ) ;Here d γ denotes the Euclidean distance on γ , and ∂γ denotes the endpoints of γ ; that is, if γ =graph h γ for h γ : I γ → T , then ∂γ = { (ˆ x, h γ (ˆ x )) : ˆ x ∈ ∂I γ } . Recall that I γ ⊂ T is always a properconnected subarc, so ∂I γ , hence ∂γ , consists of exactly two points.Additionally, let us define the following alternative of the time τ defined in § p ∈ T , wedefine ¯ τ ( p ) = 1 + max { m ≥ d ( F m − ( x, y ) , B m ) < K L − η ′ m } = min { k ≥ d ( F n − ( p ) , B n ) ≥ K L − η ′ n for all n ≥ k } . TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT27 Here, we have set η ′ = η + 12 . Clearly τ ≤ ¯ τ . A straightforward variation of the argument for Lemma 2.2 implies that ¯ τ isalmost surely finite and satisfies an analogous tail estimate to that of τ whenever P n L − η ′ n < ∞ .Precisely, we have Leb { ¯ τ > N } ≤ ∞ X n = N K L − η ′ n = O (cid:18) X n ≥ N L − η ′ n (cid:19) . (7) For the remainder of Section 6, we shall assume that the sequence { L n } is such that the right-handside of (7) is finite. The curve growth time σ S . Definition 6.2. Given p ∈ S , we define the curve growth time σ S ( p ) by σ S ( p ) = min { k ≥ ¯ τ ( p ) : Rad p k (ˆ γ k ( p k )) ≥ K L − η ′ k } , where above we write p k = F k − ( p ).In this section, we write σ = σ S for short.Our definition of σ is motivated by the following consideration. Let p ∈ S , p n = F n − ( p ), andassume σ ( p ) = n . Then, ˆ γ n ( p n ) ∩ B n = ∅ , and | I ˆ γ n ( p n ) | ≥ K L − η ′ n : this implies that F n (ˆ γ n ( p n ))is a union of approximately L η − n ≫ σ has the connotationof a mixing time : the set { σ ≤ n } ⊂ S is a region of S which has proliferated throughout T .A possible obstruction to mixing is that once this mass has proliferated, it could become ‘trapped’again by the bad sets B n . This it not possible, however, due to the way that σ is defined. Precisely,we have the following. Lemma 6.3. Let p ∈ S , and assume that σ ( p ) = n for some n ≥ . Then, Rad p k (ˆ γ k ( p k )) ≥ K L − η ′ k for all k ≥ n .Proof. It suffices to show that for any k ≥ ¯ τ ( p ), we have that Rad p k (ˆ γ k ( p k )) ≥ K L − η ′ k impliesRad p k +1 (ˆ γ k ( p k )) ≥ K L − η ′ k . This is implied directly by Lemma 2.7. (cid:3) The main result of § σ : Proposition 6.4. There is a constant C , depending only on K , M , such that the following holds.Let L be sufficiently large. Then, for any n ≥ , we have that ν { σ ( p ) > n } ≤ (cid:18) C Leb( S ) + C S (cid:19) ∞ X i = n L − η ′ i . Proposition 6.4 bears a strong resemblance to the Volume Lemma in billiard dynamics, used tocontrol the lengths of unstable manifolds; see, e.g., [11]. Remark 6.5. Let us draw a comparison between the present situation and that of a typicalnonuniformly hyperbolic system for which correlation decay and statistical properties are known,e.g., systems admitting Young towers with controllable ‘good’ return times to its base [31]. Roughlyspeaking, the typical situation is that a given ‘lump’ of mass can fail to proliferate: for example, nicehyperbolic geometry can be spoiled (as happens for Henon maps; see, e.g., [7]), or mass may become‘trapped’ somewhere (as happens for intermittent maps; see, e.g., [22]). In a typical situationadmitting a Young tower, a given ‘lump’ of mass experiences infinitely many ‘proliferations’ (returnsto the base), followed by some possibly unbounded ‘reset’ time (sojourn up the tower) before the next proliferation takes place. Thus, correlation decay estimates depend critically on the delicatebalance between these two behaviors.In contrast, the situation for our composition { F n } is simpler: at any time, some positiveproportion of ν n is ‘trapped’ in a bad region, but as time evolves, an increasingly larger proportionof the mass of ν n has ‘permanently proliferated’ throughout T .6.2.3. Proof of Proposition 6.4. We require two estimates:(A) for any p n ∈ S n , n ≥ 1, a ‘bad’ a priori estimate on Rad p n (ˆ γ n ( p n )); and(B) for Leb-almost every p ∈ S , a ‘good’ estimate for Rad p n (ˆ γ n ( p n )) for n ≫ ¯ τ ( p ) (where p n = F n − ( p ).Afterwards, we will (C) synthesize these estimates to obtain the desired estimate on the tail of σ .Let us briefly elaborate on this strategy. Before time ¯ τ ( p ), we have no control whatsoever on theorbit of p , and so our procedure may indeed produce very short curves ˆ γ n ( p n ) , p n = F n − ( p ) forsuch n . As a result, we have access to only the ‘worst possible’ estimates for Rad p n (ˆ γ n ( p n )). Wecarry these estimates out in (A) below. Once ¯ τ ( p ) has elapsed, we will leverage our control on theorbit of p after time ¯ τ ( p ) to grow the curves ˆ γ n ( p n ) to sufficient horizontal extent– this is carriedout in part (B). (A) ‘Bad’ a priori length estimate for ˆ γ n ( p n ) for all n . Here we prove the following estimate. Lemma 6.6. Let p ∈ S and write p k = F k − p for k > . Then, for any n ≥ , Rad p n (ˆ γ n ( p n )) ≥ min (cid:26) min ≤ i ≤ n (cid:26)(cid:18) n − Y j = i K L j (cid:19) − d ( p i , ∂ B i ) (cid:27) , (cid:18) n − Y j =1 K L j (cid:19) − d ( p , ∂S ) (cid:27) . Lemma 6.6 will be obtained from the corresponding identical estimate for d ( p n , ∂ P n ). Lemma 6.7. In the setting of Lemma 6.6, we have d ( p n , ∂ P n ) ≥ min (cid:26) min ≤ i ≤ n (cid:26)(cid:18) n − Y j = i K L j (cid:19) − d ( p i , ∂ B i ) (cid:27) , (cid:18) n − Y j =1 K L j (cid:19) − d ( p , ∂S ) (cid:27) . In both of Lemmas 6.6 and 6.7, the empty product Q n − j = n is to interpreted as equal to 1. Proof of Lemma 6.7. To prove this estimate, recall that for k ≥ ∂ P k = ∂ B k ∪ F k − ( ∂ P k − );thus d ( p k , ∂ P k ) = min { d ( p k , ∂ B k ) , d ( p k , F k − ( ∂ P k − ) } . Noting that Lip( F − k − ) ≤ K L k − , we obtain d ( p k , F k − ( ∂ P k − )) ≥ (2 K L k − ) − d ( p k − , ∂ P k − ) . Thus for all n ≥ a ∧ b = min { a, b } for short. d ( p n , ∂ P n ) ≥ min { d ( p n , ∂ B n ) , (2 K L n − ) − d ( p n − , ∂ P n − ) }≥ min { d ( p n , ∂ B n ) , (2 K L n − ) − d ( p n − , ∂ B n − ) , (2 K L n − ) − (2 K L n − ) − d ( p n − , ∂ P n − ) }≥ · · · ≥ d ( p n , ∂ B n ) ∧ min ≤ i ≤ n − (cid:26)(cid:18) n − Y j = i K L j (cid:19) − d ( p i , ∂ B i ) (cid:27) ∧ (cid:18) n − Y j =1 K L j (cid:19) − d ( p , ∂ P ) . The desired estimate now follows from the fact that ∂ P = ∂S ∪ ∂ B . (cid:3) TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT29 Proof of Lemma 6.6. With n ∈ N fixed, define n = max { ≤ k ≤ n − p k ∈ B k } , where we use the ad hoc convention n = 1 if p k / ∈ B k for all 1 ≤ k ≤ n − 1. Observe thatˆ γ n +1 ( p n +1 ) is formed by using Case 2 in the algorithm, and that ˆ γ k ( p k ) is formed using Case 1for every k ≥ n + 2. In particular,Rad p n (ˆ γ n +1 ( p n +1 )) ≥ d ( p n +1 , ∂ P n +1 ) , and for every n + 2 ≤ k ≤ n , we haveRad p k (ˆ γ k ( p k )) ≥ min { d ( p k , ∂ P k ) , Rad p k ( F k − (ˆ γ k − ( p k − )) } . To prove Lemma 6.6 it suffices to show thatRad p n (ˆ γ n ( p n )) ≥ min n +1 ≤ k ≤ n { d ( p k , ∂ P k ) } . (8)Once (8) is proved, Lemma 6.6 follows on plugging in the estimates for d ( p k , ∂ P k ) for 1 ≤ k ≤ n .Turning to (8): if n = n − n < n − 1, then we estimate:Rad p n (ˆ γ n ) ≥ d ( p n , ∂ P n ) ∧ L ηn − Rad p n − (ˆ γ n − ( p )) , ≥ d ( p n , ∂ P n ) ∧ L ηn − d ( p n − , ∂ P n − ) ∧ L ηn − L η ′ n − Rad p n − (ˆ γ n − ) ≥ · · ·≥ d ( p n , ∂ P n ) ∧ min n +2 ≤ i ≤ n − (cid:26)(cid:18) n − Y j = i L ηj (cid:19) d ( p i , ∂ P i ) (cid:27) ∧ Rad p n (ˆ γ n +1 ( p n +1 )) . Here we have used the simple estimateRad p j +1 F j (ˆ γ j ( p j )) ≥ L ηj Rad p j (ˆ γ j ( p j )) , (9)which follows from the expansion estimate along horizontal curves in Lemma 2.4. Replacing all L ηj terms with 1, we obtain (8). (cid:3) (B) Good length estimate for ˆ γ n ( p n ) for n ≫ τ ( p ) . Here we prove the following. Lemma 6.8. Let N ≥ , and let p ∈ S be such that ¯ τ ( p ) ≤ N < ∞ . Then for any n ≥ N , Rad p n (ˆ γ n ( p n )) ≥ min (cid:26) d ( p n , ∂ B n ) , (cid:18) n − Y k = N L ηk (cid:19) Rad p N (ˆ γ N ( p N ))) (cid:27) . (10) Proof of Lemma 6.8. The proof leans on the following claim. Claim 6.9. Let p ∈ S be such that ¯ τ ( p ) ≤ N < ∞ . Then, for all n ≥ N , we haveRad p n +1 (ˆ γ n +1 ( p n +1 )) ≥ min { d ( p n +1 , ∂ B n +1 ) , Rad p n ( F n (ˆ γ n ( p n ))) } . Proof of Claim. Observe that since n ≥ ¯ τ ( p ) ≥ τ ( p ), we always use Case 1 in the construction ofˆ γ n +1 ( p n +1 ), i.e., ˆ γ n +1 ( p n +1 ) = F n (ˆ γ n ( p n )) ∩ ∂ P n +1 ( p n +1 ). Moreover, ˆ γ n ( p n ) ⊂ P n ( p n ) by construc-tion, hence F n (ˆ γ n ( p n )) ⊂ F n ( P n ( p n )), and so we arrive atˆ γ n +1 ( p n +1 ) = F n (ˆ γ n ( p n )) ∩ B n +1 ( p n +1 ) . The desired estimate now follows. (cid:3) Fixing n ≥ N , we now estimateRad p n (ˆ γ n ( p n )) ≥ min { d ( p n , ∂ B n ) , Rad p n ( F n − (ˆ γ n − ( p n − ))) } . Observe that since d ( p n − , ∂ B n − ) ≥ K L − ηn − , it follows thatRad p n ( F n − (ˆ γ n − ( p n − ))) ≥ L ηn − Rad p n − (ˆ γ n − ( p n − )) . on applying (9). Iterating,Rad p n (ˆ γ n ( p n )) ≥ d ( p n , B n ) ∧ min N ≤ k ≤ n − (cid:26)(cid:18) n − Y i = k L ηk (cid:19) d ( p k , ∂ B k ) (cid:27) ∧ (cid:18) n − Y i = N L ηi (cid:19) Rad p N (ˆ γ N ( p N )) . Note however that for N ≤ k ≤ n − 1, we have that d ( p k , ∂ B k ) ≥ K L − η ′ k , hence L ηk · d ( p k , ∂ B k ) ≥ K L η + η ′ ) − ≫ η > / η + η ′ − > η − > 0) when L issufficiently large in terms of K , η . This yields the desired estimate. (cid:3) (C) Final estimates on the tail of σ . We are now in position to prove our estimate on Leb { p ∈ S : σ ( p ) > n } . Assume that p ∈ S and ¯ τ ( p ) ≤ n < ∞ ; finally, assume σ ( p ) > n . From Lemma 6.8it follows that Rad p n (ˆ γ n ( p n )) < K L − η ′ n · (cid:18) n − Y k = n L ηk (cid:19) − ≤ (cid:18) n − Y k = n L ηk (cid:19) − for L sufficiently large, since here we always have d ( p n , ∂ B n ) ≥ K L − η ′ n by definition of ¯ τ , σ .Plugging in our estimate from Lemma 6.6, there are two cases to consider: Case (a): For some 1 ≤ k ≤ n , we have d ( p k , ∂ B k ) < Q n − i = k K L i Q n − i = n L ηi , (again the empty product Q n − i = n is taken to equal 1) or Case (b): we have d ( p , ∂S ) < Q n − i =1 K L i Q n − i = n L ηi . By volume preservation, it follows that for each 1 ≤ k ≤ n − (cid:26) p ∈ S : τ ( p ) ≤ n , σ ( p ) > n , and Case (a) holds for value k (cid:27) ≤ C k ) · Q n − i = k K L i Q n − i = n L ηi . Additionally, using the estimate (6), we haveLeb (cid:26) p ∈ S : τ ( p ) ≤ n , σ ( p ) > n , and Case (b) holds (cid:27) ≤ Leb (cid:26) p ∈ S : d ( p, ∂S ) ≤ Q n − i =1 K L i Q n − i = n L ηi (cid:27) ≤ C S Leb( S ) Q n − i =1 K L i Q n − i = n L ηi . Thus Leb { p ∈ S : τ ( p ) ≤ n, σ ( p ) > n } ≤ (cid:0) nM + C S Leb( S ) (cid:1) Q n − i =1 K L i Q n − i = n L ηi . (11)To develop the right-hand side, observe that Q n − i =1 K L i Q n − i = n L ηi ≤ n Y i =1 K L − ηi ≤ , using that { L i } is a nondecreasing sequence, on taking L sufficiently large so that 2 K L − η ≤ i = 3 n, · · · , n − 1, we estimate: n − Y i =3 n L − ηi = (cid:18) n − Y i =3 n L − ηni (cid:19) /n ≤ n n − X i =3 n L − ηni ≤ n n − X i =3 n L − ηi TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT31 by AMGM, on noting L − ηnn < L − ηn for all n ≥ 1. Thus(11) ≤ (2 M + C S Leb( S )) n − X i =3 n L − ηi . For the final estimate, observe thatLeb { p ∈ S : σ ( p ) > n } ≤ Leb { p ∈ S : ¯ τ ( p ) ≤ n, σ ( p ) > n } + Leb { p ∈ S : ¯ τ ( p ) > n }≤ (2 M + C S ) n − X i =3 n L − ηi + 6 K M ∞ X i = n L − η ′ i ≤ (cid:0) M + C S Leb( S ) + 6 K M (cid:1) ∞ X i = n L − η ′ i on using (7) and that { L i } is nondecreasing. This completes the proof of Proposition 6.4.6.3. Disintegration of Lebesgue measure along horizontal foliation ˆ γ n . To complete ourdescription of the foliation ˆ γ n of S n , we describe here how ˆ γ n disintegrates Lebesgue measure ν n = F n − ∗ ν S = T ( S n ) Leb T | S n on S n .Below, for n ≥ γ ∈ ˆ γ n , we write ( ν n ) γ for the disintegration measure of ν n on γ ; the disintegration measures ( ν n ) γ are the (almost surely) unique family of probability measures,supported on the γ ∈ ˆ γ n , which satisfy ν n ( K ) = Z γ ∈ S n / ˆ γ n ( ν n ) γ ( γ ∩ K ) dν Tn ( γ )for Borel K ⊂ T ; here ν Tn is the pushforward of ν n onto the quotient space of equivalence classes S n / ˆ γ n . Lemma 6.10. Let n ≥ and fix γ ∈ ˆ γ n . Let ρ nγ denote the density of ( ν n ) γ with respect to Leb γ .Then, for any p, q ∈ γ , we have that ρ nγ ( p ) ρ nγ ( q ) = det( dF n − n +1 | T γ n ) ◦ ( F n − n +1 ) − ( q )det( dF n − n +1 | T γ n ) ◦ ( F n − n +1 ) − ( p ) Here n = max (cid:0) { } ∪ { ≤ k ≤ n − p k ∈ B k } (cid:1) , p n ∈ γ is an (arbitrary) representative and p k ∈ S k is such that F n − k p k = p n for each k ≤ n , and γ n +1 is the atom in ˆ γ n +1 for which F n − n +1 ( γ n +1 ) ⊃ γ .Proof. To start let us describe the disintegration measures ( ν ) ˆ γ ( p ) for p ∈ S . It is clear that( ν ) ˆ γ ( p ) = 1Leb H p (ˆ γ ( p )) Leb H p | ˆ γ ( p ) , (12)where H p is as in § γ ) denotes the arc length of a smooth connected curve γ ⊂ T .Thus Lemma 6.10 holds trivially in this case with n = 1.Inductively, let us express the disintegration ν n +1 in terms of that for ν n . Observe that ν n +1 = ( F n ) ∗ ν n | S n ∩ B n + ( F n ) ∗ ν n | S n \ B n ;since S n ∩ B n , S n \ B n ∈ P n it suffices to consider these separately in working out the disintegrationmeasures ( ν n +1 ) γ , γ ∈ ˆ γ n +1 .On F n ( S n ∩ B n ), Case 2 is applied in constructing ˆ γ n +1 | F n ( S n ∩ B n ) , and so disintegration measuresare obtained using the analogue of (12) with n + 1 replacing 1.On F n ( S n \ B n ), we apply Case 1 in the construction of ˆ γ n +1 , i.e., ˆ γ n +1 = P n +1 | F n ( S n ∩ B n ) ∨ F n (ˆ γ n | S n ∩ B n ). In particular, the disintegration ( ν n +1 | F n ( S n \ B n ) ) γ , γ ∈ ˆ γ n +1 can be obtained by disintegrating, for each ˇ γ ∈ ˆ γ n , the measures ( F n ) ∗ (cid:0) ( ν n ) ˇ γ (cid:1) against the (finite) partition P n +1 | F n (ˇ γ ) .To wit, if γ ∈ ˆ γ n +1 | F n ( S n \ B n ) has γ ⊂ F n (ˇ γ ) for ˇ γ ∈ ˆ γ n +1 , then( ν n +1 ) γ = 1( ν n ) ˇ γ ( F − n γ ) ( F n ) ∗ (cid:0) ( ν n ) ˇ γ (cid:1) | γ . In particular, we have shown that for any p, q ∈ γ , we have that ρ n +1 γ ( p ) ρ n +1 γ ( q ) = det( dF n | T ˇ γ ) ◦ F − n ( q )det( dF n | T ˇ γ ) ◦ F − n ( p ) · ρ n ˇ γ ◦ F − n ( p ) ρ n ˇ γ ◦ F − n ( q ) . Lemma 6.10 follows by iterating the above relations from n + 1 to n − (cid:3) Description of ( F n ) ∗ ν S . Here we synthesize the results of § F n ) ∗ ν S as foliated by a collection of fully crossing horizontalcurves with controlled disintegration densities. Proposition 6.11. Let n ≥ . Then, there is a measurable set G ⊂ F n S and a measurable partition G of G with the following properties. (a) Each atom γ ∈ G is of the form graph h γ where h γ : (0 , → T is a C , fully crossinghorizontal curve with k h ′ γ k C = O ( L − ηn ) . (b) We have the estimate ν n +1 ( G ) ≥ − O ( L − (1 − η ) n ) − ν S { σ > n }≥ − (cid:18) O (1) + C S + C Leb( S ) (cid:19) ∞ X i = ⌊ n/ ⌋ L − (1 − η ) i (13) on plugging in the estimate in Proposition 6.4. (c) Let ν G denote the restriction ν n +1 | G and let { ( ν G ) γ } γ ∈G denote the canonical disintegrationof ν G with respect to G by probability measures supported on each γ ∈ G . Let ρ γ : γ → [0 , ∞ ) denote the density of ( ν G ) γ with respect to Leb γ . Then for any p , p ∈ γ we have ρ ( p ) ρ ( p ) ≤ e CL − ηn . Proof. To start, defineˆ G n = { ˆ γ ∈ ˆ γ n : ( ν n ) ˆ γ F n − { σ ≤ n } > } and ˆ G n = [ ˆ γ ∈ ˆ G n ˆ γ . By Lemma 7.2 in the appendix, we have ν n ( ˆ G n ) = ν Tn { ˆ γ ∈ ˆ G n } ≥ ν { σ ≤ n } . Recalling the notation in Lemma 2.7, we define G by G = [ ˆ γ ∈ ˆ G n ¯Γ n (ˆ γ ) and G = [ γ ∈G γ = [ ˆ γ ∈ ˆ G n F n (ˆ γ \ B n (ˆ γ )) , noting that G partitions G into horizontal curves γ which satisfy item (a) by construction.To check item (b), for each ˆ γ ∈ ˆ γ n and subset K ⊂ ˆ γ we have that( ν n ) ˆ γ ( K ) ≤ C Len(ˆ γ ) Leb ˆ γ ( K ) , TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT33 on applying the distortion estimate in Lemma 2.5 to the density ρ n ˆ γ derived in Lemma 6.10. SinceLen(ˆ γ ) − = O ( L − η ′ n ) from the fact that ˆ γ ∩ F n − { σ ≤ n } 6 = ∅ , we obtain the estimate( ν n ) ˆ γ ( B n (ˆ γ )) = O (cid:18) L − ηn L − η ′ n (cid:19) = O ( L − (1 − η ) n )on plugging in K = B n (ˆ γ ). Thus (13) follows on noting − η ′ = − η = η − .For item (c), let p , p ∈ γ for some γ ∈ G , and assume that γ ∈ ¯Γ n (ˆ γ ) for ˆ γ ∈ ˆ γ n . Then, ρ γ ( p ) ρ γ ( p ) = det( dF n | T ˆ γ ) ◦ F − n ( p )det( dF n | T ˆ γ ) ◦ F − n ( p ) · ρ n ˆ γ ◦ F − n ( p ) ρ n ˆ γ ◦ F − n ( p )in the notation of § ≤ e CL − ηn k p − p k by Lemma 2.5. For the secondfactor, note that k F − n ( p ) − F − n ( p ) k ≤ L − ηn k p − p k by Lemma 2.4, and so Lemma 6.10 yieldsthe estimate ≤ e CL − ηn · L − ηn k p − p k ≤ e CL − ηn . The estimate in item (c) follows. (cid:3) Decay of correlations estimates Leaning on the mixing mechanism explored in the previous section, we complete here the proofof Theorem C.In § ϕ is the characteristic functionof a small square (Proposition 7.1). In § § S is a small square andgive the proof of Proposition 7.1. We assume throughout § η ∈ (1 / , has been fixed, and that { L n } has the property that P n L − η ′ n < ∞ , where η ′ = η +12 is as in § Reduction. We will show here that to prove Theorem C, it suffices to prove the following. Proposition 7.1. Let R be a square in T of side length ℓ , and let ν denote the normalized Lebesguemeasure restricted to R . Let ψ : T → R be α -Holder continuous. Then (cid:12)(cid:12)(cid:12)(cid:12) Z ψ ◦ F n dν − Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k α max (cid:26) L − min { η − ,α (1 − η ) / ( α +2) }⌊ n/ ⌋ , ℓ − ∞ X i = ⌊ n/ ⌋ L − (1 − η ) i (cid:27) . Proof of Theorem C assuming Proposition 7.1. Below, n ≥ α -Holder continuous ϕ, ψ : T → R . Let us write S n for the first element in the max {· · · } in Proposition 7.1 andwrite T n for the summation in the second term, so that the bound on the right-hand side reads as ≤ C k ψ k α max {S n , ℓ − T n } .With K ∈ N to be specified later, subdivide T into rectangles R i,j , ≤ i, j ≤ K of side length ℓ = 1 /K each. We set ϕ i,j = inf p ∈ R i,j ϕ ( p ) . Define ˆ ϕ := P Ki,j =1 ϕ i,j χ R i,j , so that Z (cid:0) ϕ − ˆ ϕ (cid:1) d Leb = O ( k ϕ k α · ℓ α ) . Let ν i,j denote normalized Lebesgue measure on R i,j . Then Z ψ ◦ F n · ϕ = Z ψ ◦ F n · (cid:0) ϕ − ˆ ϕ (cid:1) + K X i,j =1 ℓ ϕ i,j Z ψ dF n ∗ ν i,j . For the first term, Z ψ ◦ F n · ( ϕ − ˆ ϕ ) = O ( k ψ k α k ϕ k α ℓ α ) . Similarly, we estimate Z ψ · Z ϕ = Z ( ϕ − ϕ k ) · Z ψ + K X i,j =1 ℓ ϕ i,j Z ψ = O ( k ψ k α k ϕ k α ℓ α ) + K X i,j =1 ℓ ϕ i,j Z ψ hence (cid:12)(cid:12)(cid:12)(cid:12) Z ψ ◦ F n · ψ − Z ψ Z ϕ (cid:12)(cid:12)(cid:12)(cid:12) ≤ K X i,j =1 ℓ ϕ i,j (cid:12)(cid:12)(cid:12)(cid:12) Z ψ dF n ∗ ν i,j − Z ψ (cid:12)(cid:12)(cid:12)(cid:12) + O ( k ψ k C [ ϕ ] α ℓ α )= k ψ k α k ϕ k α · O (cid:0) S n + ℓ − T n + [ ϕ ] α ℓ α (cid:1) . Setting K = (cid:22)(cid:18) [ ϕ ] α T n (cid:19) / (2+ α ) (cid:23) we obtain the estimate (cid:12)(cid:12)(cid:12)(cid:12) Z ψ ◦ F n · ϕ − Z ϕ Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k α k ϕ k α α α ( T α α n + S n ) . The only difference between this and our desired estimate is the exponent of k ϕ k α on the right-hand side. To fix this, define ˇ ϕ = ϕ/ k ϕ k α and note k ˇ ϕ k α = 1; for this function we have (cid:12)(cid:12)(cid:12)(cid:12) Z ψ ◦ F n · ˇ ϕ − Z ˇ ϕ Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k α ( T α α n + S n ) , and so the desired estimate follows on multiplying both sides by k ϕ k α . To complete the proof,observe that max {S n , T αα +2 n } ≤ max (cid:26) L − η ⌊ n/ ⌋ , (cid:18) X i = ⌊ n/ ⌋ L − (1 − η ) i (cid:19) αα +2 (cid:27) since T αα +2 n always dominates L − α (1 − η ) / ( α +2) ⌊ n/ ⌋ . (cid:3) Proof of Proposition 7.1. To complete the proof of Theorem C, it remains to prove Propo-sition 7.1. We combine the description in Proposition 6.11 of the foliation by long horizontal curveswith the mixing estimate in Proposition 2.8 along those horizontal curves.To wit: let ψ : T → R be α -Holder continuous and let R be a square of side length ℓ as inthe statement of Proposition 7.1. With ν denoting the Lebesgue measure restricted to R , and (fornotational convenience) appling the substitution n n , we will estimate Z ψ ◦ F n dν = Z ψ ◦ F nn +1 d ( F n ∗ ν ) . (14)For each k ≥ ν k = F k − ∗ ν , where ν = ν . Applying Proposition 6.11 to S = R , we obtainthe collection G of horizontal curves foliating the set G ⊂ F n R . In the notation of Proposition 6.4, TATISTICAL PROPERTIES FOR COMPOSITIONS OF STANDARD MAPS WITH INCREASING COEFFICIENT35 we have C R = O ( ℓ − ), and so ν n +1 ( G c ) = O (cid:18) ℓ − ∞ X i = ⌊ n/ ⌋ L − (1+ η ) i (cid:19) . Returning to the estimate of (14),(14) = O ( k ψ k α ν n +1 ( G c )) + Z ψ ◦ F nn +1 dν G = O ( k ψ k α ν n +1 ( G c )) + Z G/ G (cid:18) Z γ ψ ◦ F nn +1 d ( ν G ) γ (cid:19) dν TG , where the transversal measure ν TG is the pushforward of ν G onto G/ G .Fixing γ ∈ G , we have by the density estimate in Proposition 6.11 that Z γ ψ ◦ F nn +1 d ( ν G ) γ = (1 + O ( L − ηn )) Z γ ψ ◦ F nn +1 d Leb γ , and so applying Proposition 2.8 with m n + 1 , n n , we have Z γ ψ ◦ F nn +1 d ( ν G ) γ = (1 + O ( L − ηn )) Len( γ ) · Z ψ + (1 + O ( L − ηn )) k ψ k α · O (cid:18) L − α (1 − η ) / (2+ α )2 n + L − ηn +1 + n − X k = n +1 L − ηk (cid:19) = Z ψ + k ψ k α · O (cid:18) L − α (1 − η ) / (2+ α )2 n + L − ηn + n − X k = n +1 L − ηk (cid:19) . Collecting these estimates, we conclude (cid:12)(cid:12)(cid:12)(cid:12) Z ψ ◦ F n dν − Z ψ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ψ k α (cid:18) L − min { η − ,α (1 − η ) / (2+ α ) } n + ℓ − ∞ X i = ⌊ n/ ⌋ L − (1 − η ) i (cid:19) . 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