Quantitative global-local mixing for accessible skew products
aa r X i v : . [ m a t h . D S ] J un QUANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEWPRODUCTS
PAOLO GIULIETTI, ANDY HAMMERLINDL AND DAVIDE RAVOTTIA
BSTRACT . We study global-local mixing for accessible skew products with amixing base. For a dense set of almost periodic global observables, we proverapid mixing; and for a dense set of global observables vanishing at infinity, weprove polynomial mixing. More generally, we relate the speed of mixing tothe “low frequency behaviour” of the spectral measure associated to our globalobservables. Our strategy relies on a careful choice of the spaces of observablesand on the study of a family of twisted transfer operators. C ONTENTS
1. Introduction 12. Setup and main results 33. The main result 54. Skew-products over one-sided subshifts 105. Cancellations for twisted transfer operators 136. Contraction 177. Rapid decay 208. Proof of Theorem 4.5 229. From accessibility to collapsed accessibility 2410. Proof of Theorem 3.8 25Appendix A. Accessibility and symbolic dynamics 27Appendix B. Proofs of Lemmas 3.2, 3.5, 3.7, and 3.1 29Appendix C. Proofs of Lemma 10.2 and Proposition 10.3 31References 351. I
NTRODUCTION
Dynamical systems within the category of skew products have a long history.They receive attention for many different reasons: in earlier ergodic theory, theywere studied as mild generalizations of suspensions which cannot be factored [12,Chapter 10], providing examples of simple partially hyperbolic systems; in recentdays, they are often used to model real life situations [18, 2]. As their name suggest,
Mathematics Subject Classification.
Primary: 37A40, 37A25 ; Secondary: 37D30, 37B10 .
Key words and phrases. accessible systems, skew products, decay of correlations, symbolic dy-namics, global observables, global-local mixing, infinite measure.P.G. Centro di Ricerca Matematica Ennio de Giorgi, Scuola Normale Superiore, Piazza dei Ca-valieri 7, 56126 Pisa, Italy. E-mail: [email protected] .A.H. School of Mathematics, Monash University, Victoria 3800 Australia.E-mail: [email protected] .D.R. School of Mathematics, Monash University, Victoria 3800 Australia.E-mail: [email protected] . they are products of a base dynamics and a fiber dynamics. Here, we are concernedwith the statistical properties of these systems.One of the simplest examples of a partially hyperbolic skew product is givenby circle extensions of Anosov diffeomorphisms on the 2-dimensional torus T , de-fined as follows. Given an Anosov diffeomorphism A : T → T and a smoothmap f : T → T , a skew product F : T → T over A induced by f is definedby F ( x , r ) = ( Ax , r + f ( x )) . The torus T is equipped with a product measure µ u × Leb, where µ u is any Gibbs measure with Hölder potential u and Leb is theLebesgue measure on the fibers . In this case, Dolgopyat [13] proved that genericfunctions f induce skew products F with rapid decay of correlations , or rapidmixing , i.e. decorrelation of C ∞ -observables is faster than any given polynomial.The speed of mixing may, in fact, be exponential, but this is still an open problem([13, Problem 2]). Dolgopyat’s result holds in general for compact group exten-sions. The interested reader can also check the introductions of [26, 7, 17] for anoverview of old and new results on skew products.In this paper, we are interested in skew products with non-compact fibers. Inparticular we will consider R -extensions of topologically mixing Anosov diffeo-morphisms. For an introduction to infinite ergodic theory, we refer the reader toAAronson’s book [1]. In our setting, Guivarc’h showed that any Hölder function f with zero integral which is not cohomologous to a constant induces an ergodicskew product [20] (see also [11]).Concerning stronger statistical properties, an historical perspective on the vari-ous possible definitions of mixing can be found in [23]. We will be interested in global-local mixing , a notion introduced by Lenci in [23], namely we will studythe correlations between global and local observables (see Definition 2.1). Localobservables are akin to compactly supported observables, while global observablesare supported over most of the of phase space. One possible concern about thenotion of global-local mixing may be the seemingly arbitrary choice of the aver-aging involved in the definition of global observables. In our setting, the infinitevolume average is analogous to statistical infinite volume limits introduced andrefined along the years by Van Hove, Fisher [25, Section 3.3] and Ruelle [29, Sec-tion 3.9], which built on the inspirational work of Bogoliubov [4]. Global-localmixing has been studied in different situations, for example random walks [23, 24],mechanical systems [15, 14] and one dimensional parabolic systems [8].The main ingredient needed to study our class of skew product is accessibility (see, among others, [10, 9, 30]). The skew-product F is accessible if, roughlyspeaking, it is possible to reach any point in the space by moving along segmentsof stable and unstable manifolds.From the measure-theoretic point of view, using Markov partitions, we can trans-late the problem to study a skew product over a subshift of finite type, keeping thesame R -fibers. The question whether accessibility is preserved passing to symbolicdynamics appears to be delicate, and will be discussed in Appendix A. Our mainresult, Theorem 3.8, provides quantitative estimates for the decay of correlationsof global and local observables for an accessible R -extension of a symbolic shift.To the best of our knowledge, this is the first quantitative result in the context ofglobal-local mixing.Contrary to the case of compact group extensions, we cannot expect exponentialmixing in general, since, taking Fourier transforms, we have to deal with arbitrary Recall that if one chooses the potential as the det ( DA ) | E s , one recovers the usual SRB measure. UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 3 low frequencies. Indeed, we will show in Theorem 3.8 that the speed of conver-gence of correlations depends on the behavior near zero of the spectral measureassociated to the global observable (namely, its inverse Fourier-Stieltjes transform):if the support of this measure intersects a neighbourhood of 0 only at 0, then mix-ing is rapid (Theorem 2.2) as we expect from Dolgopyat’s result; in other cases weobtain polynomial estimates (Theorem 2.3, Theorem 2.4), which correspond to theexpected behaviour (see Remark 2.5). Note that since the infinite volume averageequals the value of the associated spectral measure at zero, our choice of infinitevolume average is natural.The main novelties of the work rely on a careful choice of the functional spacesinvolved. The accessibility hypothesis, when coupled with a standard central limittheorem for the underlying symbolic dynamics on the basis, allows for transferoperator bounds on suitable splitting of lower and higher frequency modes and theexploiting of cancellation effects due to accessibility.1.1.
Outline of the paper.
The rest of the paper is organized as follows. In Section2 we rigorously introduce our framework and state our main results. In Section 3,we describe in details the classes of global and local observables we consider andwe state our core result, Theorem 3.8. We then deduce Theorems 2.2, 2.3 and 2.4from Theorem 3.8.In Section 4, we present a preliminary result in the non-invertible case of skewproducts over one-sided subshifts, Theorem 4.5. We also describe a “collapsed ac-cessibility” property, which constitutes the main working assumption on the skewproduct in this setting.The main tool to prove Theorem 4.5 is a family of twisted transfer operators. InSection 5, we show how the collapsed accessibility property can be exploited toobtain some cancellations in the expression for the twisted transfer operators, as inthe work of Dolgopyat [13]. In Section 6, we prove some estimates on the normof the twisted transfer operators. For large twisting paramenters, the estimates areobtained exploiting the results in Section 5; for small parameters, we apply somestandard results in the theory of analytic perturbations of bounded linear operators.Section 7 contains some technical results that will be applied to prove the maintheorems. Section 8 is devoted to the proof of Theorem 4.5. In order to deduceTheorem 3.8 from Theorem 4.5, in Section 9 we deduce the collapsed accessibil-ity property for a one-sided skew-product from the accessibility property of thecorresponding two-sided skew-product. In Section 10, we prove Theorem 3.8.In Appendix A, we discuss the problem whether the accessibility property for askew product over an Anosov diffeomorphism is equivalent to the accessibility ofthe associated symbolic system. Appendices B and C contain the proofs of severaltechnical results. 2. S
ETUP AND MAIN RESULTS
Let σ : Σ → Σ be a topologically mixing two-sided subshift of finite type, equippedwith a Gibbs measure µ = µ u with respect to a Hölder potential u (see, e.g., [6, §1]or [26, §3]). For 0 < θ <
1, define the distance d θ ( x , y ) = θ max { j ∈ N : x i = y i for all | i | < j } . Let us denote by F θ the space of Lipschitz continuous functions w : Σ → C equippedwith the norm(1) k w k θ = k w k ∞ + | w | θ , where | w | θ = sup x = y | w ( x ) − w ( y ) | d θ ( x , y ) . PAOLO GIULIETTI, ANDY HAMMERLINDL, AND DAVIDE RAVOTTI
We consider the skew-product(2) F : Σ × R → Σ × R , F ( x , r ) = ( σ x , r + f ( x )) , where f : Σ → R is a Lipschitz continuous function with zero average, R Σ f d µ u = F is accessible , see Sections 3 and 9 for definitions.We are interested in the mixing properties of the map F with respect to the infinite measure ν = µ × Leb, where Leb is the Lebesgue measure on R . Definition 2.1. A local observable is any function ψ ∈ L ( ν ) . A global observable is any function Φ ∈ L ∞ ( ν ) such that the following limit exists (3) ν av ( Φ ) : = lim R → ∞ R Z Σ × [ − R , R ] Φ ( x , r ) d ν ( x , r ) . If ψ is a local observable, we will write ν ( ψ ) = R Σ × R ψ d ν . We will show inLemma 3.2 below that if Φ is a global observable, then so is Φ ◦ F , and the average ν av defined in (3) is invariant under F .For any pair of global and local observables ( Φ , ψ ) , let us denote by cov ( Φ , ψ ) the covariance cov ( Φ , ψ ) : = ν ( Φ · ψ ) − ν av ( Φ ) ν ( ψ ) . We are interested in showing “ global-local mixing ”, namely in proving that thecorrelations cov ( Φ ◦ F n , ψ ) satisfy(4) lim n → ∞ cov ( Φ ◦ F n , ψ ) = . and the rate of convergence to such limit (also known as the rate of decay of corre-lations).The main result of this paper, Theorem 3.8 below, establishes quantitative global-local mixing estimates . Since some preliminary work is needed, the statement ofthe main theorem is postponed to Section 3. We state here some corollaries whichshould give the reader a rather complete picture of the possible scenarios. Theorem2.2 states that, for a dense class of almost periodic global observables, we have rapid mixing , namely the decay of correlations is faster than any given polynomial,in analogy to Dolgopyat’s result [13] in the case of circle extensions. On the otherhand, for a dense class of global observables which vanish at infinity, Theorems 2.3and 2.4 state that the decay is polynomial. The bound in Theorem 2.4 is generically optimal , as we show in §3.6.Let us fix some notation. Let C k ( R ) be the space of k -times differentiable func-tions on R . We will denote by P the subspace of C ( R ) consisting of 2 π -periodicfunctions; by C ( R ) the space of continuous functions on R which vanish at infin-ity, and by C ∞ c ( R ) the subspace of infinitely differentiable functions with compactsupport. The space C ∞ c ( R ) has the structure of a Fréchet space, induced by thefamily of seminorms k · k C k , for k ∈ N . We will say that a map ψ : Σ → C ∞ c ( R ) isLipschitz if it is Lipschitz with respect to k · k C k , for all k ∈ N .In the rest of the paper, we will implicitely identify maps a from Σ to some spaceof complex-valued measurable functions over R with complex-valued measurablefunctions on Σ × R by setting a ( x , r ) = [ a ( x )]( r ) . Theorem 2.2 (Rapid global-local mixing) . Assume that F, defined as in (2) , isaccessible. For any Lipschitz map ψ : Σ → C ∞ c ( R ) , for any Lipschitz map Φ : Σ → We recall that the space of almost periodic functions is the closure of the space of trigonometricpolynomials with respect to the uniform norm.
UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 5 P , and for every ℓ ∈ N , there exists a constant C = C ( ℓ, ψ , Φ ) ≥ such that forall n ∈ N , | cov ( Φ ◦ F n , ψ ) | ≤ Cn − ℓ . The situation is different when the global observable vanishes at infinity. Notethat, in this case, the average ν av defined in (3) is zero. Theorem 2.3 (Polynomial global-local mixing I) . Assume that F, defined as in (2) ,is accessible. There exists a space of bounded continuous functions D ⊂ C ( R ) ,which is dense in C ( R ) with respect to k · k ∞ , and there exists α > such thatthe following holds. For any Lipschitz map ψ : Σ → C ∞ c ( R ) , and for any map Φ : Σ → D which satisfies some explicit Lipschitz condition, there exists a constantC = C ( ψ , Φ ) ≥ such that for all n ∈ N , | cov ( Φ ◦ F n , ψ ) | ≤ Cn − α . The Lipschitz conditions for the global observable in Theorem 2.3 will be statedexplicitly in Section 3 below, as well as a bound on α . If we further assume thatthe global observable takes values in W ( R ) , where W ( R ) is the Sobolev space of L functions with weak derivative in L , then the statement reads as follows. Theorem 2.4 (Polynomial global-local mixing II) . For any Lipschitz map ψ : Σ → C ∞ c ( R ) , for any Lipschitz map Φ : Σ → W ( R ) , and for any ε > , there exists aconstant C = C ( ψ , Φ , ε ) ≥ such that for all n ∈ N , | cov ( Φ ◦ F n , ψ ) | ≤ Cn − + ε . If moreover Φ : Σ → W ( R ) ∩ L p ( R ) for some ≤ p ≤ , then | cov ( Φ ◦ F n , ψ ) | ≤ Cn − p + ε . Remark 2.5.
The bound in Theorem is optimal: we will provide an example in§ of a pair of global and local observables Φ , ψ for which the correlations arebounded below by | cov ( Φ ◦ F n , ψ ) | ≥ Bn − for some constant B > , and, on theother hand, Theorem implies that for any ε > there exists a constant C > such that | cov ( Φ ◦ F n , ψ ) | ≤ Cn − + ε .
3. T
HE MAIN RESULT
In this section, we first recall the definition of accessibility for F as in (2); then,we describe the classes of global and local observables we consider, and we stateour main result. The proofs of Lemmas 3.2, 3.5, 3.7 and 3.1 are contained in theAppendix B. Finally, we deduce Theorems 2.2, 2.3 and 2.4 from Theorem 3.8.3.1. Accessibility.
For each point x ∈ Σ , we define the stable and unstable set at x by, respectively, W s ( x ) = { y ∈ Σ : there exists n ∈ Z such that y i = x i for all i ≥ n } , W u ( x ) = { y ∈ Σ : there exists n ∈ Z such that y i = x i for all i ≤ n } . By definition, for any y ∈ W s ( x ) , d θ ( σ n x , σ n y ) → y ∈ W u ( x ) , d θ ( σ − n x , σ − n y ) →
0; moreover, note that d θ attains a discreteset of values { θ i } i ∈ N .The skew product (2) is partially hyperbolic in the following sense. Let usdenote by f n ( x ) = ∑ n − i = f ◦ σ i ( x ) the n -th Birkhoff sum at x . Let us define for any PAOLO GIULIETTI, ANDY HAMMERLINDL, AND DAVIDE RAVOTTI ( x , r ) ∈ Σ × R W s ( x , r ) = { ( y , s ) ∈ Σ × R : y ∈ W s ( x ) and s − r = lim n → ∞ f n ( x ) − f n ( y ) } , W u ( x , r ) = { ( y , s ) ∈ Σ × R : y ∈ W u ( x ) and s − r = lim n → ∞ f n ( σ − n y ) − f n ( σ − n x ) } . We equip Σ × R with the product distance given bydist (cid:0) ( x , r ) , ( y , s ) (cid:1) = d θ ( x , y ) + | s − t | ;it is easy to see thatlim n → ∞ dist ( F n ( x , r ) , F n ( y , s )) = ( y , s ) ∈ W s ( x , r ) , lim n →− ∞ dist ( F n ( x , r ) , F n ( y , s )) = ( y , s ) ∈ W u ( x , r ) . The sets W s ( x , r ) and W u ( x , r ) are called the (strong) stable and (strong) unstablemanifold at ( x , r ) ∈ Σ × R . Vertical lines { x } × R constitute the center manifolds ,namely they form an invariant fibration and the action of F on each line is isometric.We now define the accessibility property. A su -path from ( x , r ) to ( y , s ) is a finitesequence ( x i , r i ) ∈ Σ × R , for 0 ≤ i ≤ m for some m ∈ N , such that ( x , r ) = ( x , r ) , ( x m , r m ) = ( y , s ) , and ( x i , r i ) ∈ W s ( x i − , r i − ) or ( x i , r i ) ∈ W u ( x i − , r i − ) for all 1 ≤ i ≤ m . We say that F is accessible if for any two points ( x , r ) , ( y , s ) ∈ Σ × R thereis a su -path from ( x , r ) to ( y , s ) .A consequence of the accessibility property is the following fact, which will beproved in Appendix B. Lemma 3.1.
If F is accessible, then f is not cohomologous to zero.
The classes of global and local observables.
We now describe the classes ofglobal and local observables we consider. Let us start by observing that the averagedefined in (3) is invariant under F . Lemma 3.2. If Φ is a global observable according to Definition , then Φ ◦ F isa global observable and ν av ( Φ ◦ F ) = ν av ( Φ ) . We will denote by S the Fréchet space of Schwartz functions on R , with thefamily of seminorms k g k a ,ℓ : = sup r ∈ R | r | a (cid:12)(cid:12)(cid:12)(cid:12) d ℓ ( d r ) ℓ g ( r ) (cid:12)(cid:12)(cid:12)(cid:12) . We will say that a function ψ : Σ → S is Lipschitz if it is Lipschitz with respectto k · k a ,ℓ for all a , ℓ ∈ N . Starting from definition (2.1), we restrict ourselves fromnow on to smaller classes of observables. Definition 3.3 (Local observables) . We denote by L ⊂ L ( ν ) the space of Lipschitzfunctions ψ : Σ → S . Let η be a complex measure over R . We will denote by | η | the variation of η and by k η k TV = | η | ( R ) its total variation . We recall that the Fourier-Stieltjestransform b η ( r ) of a complex measure η of finite total variation is the L ∞ functiondefined by b η ( r ) : = Z R e − ir ξ d η ( ξ ) . The space A of all Fourier-Stieltjes transforms is an algebra of functions, calledthe Fourier-Stieltjes algebra . We equip A with the total variation norm, namely,for b η , b η ∈ A , we set k b η − b η k : = k η − η k TV . UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 7
Definition 3.4 (Global observables) . We denote by G ⊂ L ∞ ( ν ) the space of Lips-chitz functions Φ : Σ → A which satisfy the follwing tightness condition: there exist a , A > such that for all r ≥ and x ∈ Σ we have | η x | ( R \ [ − r , r ]) ≤ Ar − a , (TC) where b η x = Φ ( x ) . The tightness condition (TC) will be exploited in the proofs of Lemma 10.2 andProposition 10.3 in Appendix C to ensure some compactness property. The follow-ing lemma shows that elements of G are indeed global observables; more precisely,the average ν av of Φ is the average of the values of the associated measures η x at 0. Lemma 3.5. If Φ ∈ G , then Φ is a global observable according to Definition and ν av ( Φ ) = Z Σ η x ( { } ) d µ ( x ) , where, as before, b η x = Φ ( x ) . We conclude this section by providing a useful criterion to determine whether agiven function is the Fourier-Stieltjes transform of a finite complex measure whichsatisfies (TC). A positive definite function is any function g : R → C such that n ∑ i , j = g ( x i − x j ) z i z j ≥ , for all n ≥ x i , x j ∈ R and z i , z j ∈ C . By Bochner’s theorem, a function g iscontinuous and positive definite if and only if it is the Fourier-Stieltjes transform b η of a finite positive measure η on R (see, e.g., [28, Theorem IX.9]). For example, itis easy to check that g ( x ) = e ix or g ( x ) = cos ( x ) are positive definite functions. Aless trivial example is the function g ( x ) = | x | + ; the fact that g is positive definitefollows from Pólya’s Criterion: any positive, continuous, even function which, forpositive x , is non-increasing, convex and tends to 0 for x → ∞ is the Fourier-Stieltjestransform of an L function, thus positive definite. Lemma 3.6.
Any linear combination of Lipschitz positive definite functions is theFourier-Stieltjes transform of a complex measure of finite total variation whichsatisfies (TC) . Since any complex measure of finite total variation is a linear combination ofpositive finite measures, the proof of the lemma above follows immediately fromthe following tail estimate, whose proof can be found in Appendix B.
Lemma 3.7.
Let η be a finite positive measure on R , let Φ ( r ) be its Fourier-Stieltjes transform. Then, if Φ ( r ) is Lipschitz of constant L, for all r > we have η ( R \ [ − r , r ]) ≤ Lr . Statement of the main result.
For any Φ ∈ G and for any r >
0, let us definethe “low frequency variation” as(5) LF ( Φ , r ) : = Z Σ | η x | (cid:0) ( − r , r ) \ { } (cid:1) d µ ( x ) . Notice that LF ( Φ , · ) is monotone and LF ( Φ , r ) → r →
0. We are now readyto state our main result.
PAOLO GIULIETTI, ANDY HAMMERLINDL, AND DAVIDE RAVOTTI
Theorem 3.8 (Quantitative global-local mixing) . Assume that F, defined as in (2) ,is accessible. Then, for every ψ ∈ L , for every Φ ∈ G , for any k ∈ N , and for every ε > , there exists a constant C = C ( Φ , ψ , k , ε ) > such that for every n ∈ N , | cov ( Φ ◦ F n , ψ ) | ≤ C (cid:16) LF (cid:16) Φ , n − + ε (cid:17) + n − k (cid:17) . The bound in Theorem 3.8 is the sum of two terms, namely a superpolynomialterm and the contribution given by the measures | η x | close to 0. In particular, if thesupport of the measures η x does not intersect some neighbourhood of 0, then thedecay of correlations is superpolynomial. On the other hand, for example underthe assumptions of Theorem 2.4, the measures | η x | are absolutely continuous andthe decay is polynomial.In the rest of the section we prove Theorems 2.2, 2.3 and 2.4 from the resultabove.3.4. Proof of Theorem 2.2.
We deduce Theorem 2.2 from Theorem 3.8.By the theory of Fourier series, any p ∈ P ⊂ C ( R ) , by periodicity, is theFourier-Stieltjes transform of a discrete measure η of the form η = ∑ n ∈ Z a n δ n ,where a n ∈ C and δ n is the Dirac measure at n . We claim that any Lipschitz map Φ : Σ → P is contained in G . Theorem 3.8 then immediately implies the result,since | η x | (( − , ) \ { } ) = Φ ( x ) = b η x ).We first check the Lipschitz condition. For x ∈ Σ , let us write η x = ∑ n ∈ Z a n ( x ) δ n .Since Φ ( x ) ∈ C ( R ) , it follows that lim n → ∞ | n a n ( x ) | =
0. In particular, the se-quence | a n ( x ) | · ( + i | n | ) is square-summable (notice that ina n ( x ) are the Fouriercoefficients of the derivative Φ ( x ) ′ ). Thus, for any x , y ∈ Σ , by Cauchy-Schwartz,we have k Φ ( x ) − Φ ( y ) k = k η x − η y k TV = ∑ n ∈ Z | a n ( x ) − a n ( y ) | = ∑ n ∈ Z | a n ( x ) − a n ( y ) | · + i | n | + i | n |≤ ∑ n ∈ Z + n ! · ∑ n ∈ Z | a n ( x ) − a n ( y ) | + | a n ( x ) − a n ( y ) | · n ! . Hence, by Plancharel formula, there exists a constant C > k Φ ( x ) − Φ ( y ) k ≤ C (cid:0) k Φ ( x ) − Φ ( y ) k ∞ + k Φ ( x ) ′ − Φ ( y ) ′ k ∞ (cid:1) ≤ C k Φ ( x ) − Φ ( y ) k C . This shows that Φ : Σ → G is Lipschitz.We now show that Φ satisfies the tightness condition (TC). Since Φ ( x ) ∈ C ( R ) ,we can bound | a n ( x ) | ≤ k Φ ( x ) ′′ k ∞ n − . Thus, for any r ≥ | η x | ( R \ [ − r , r ]) = ∑ | n | > r | a n ( x ) | ≤ k Φ ( x ) ′′ k ∞ ∑ | n | > r n − ≤ k Φ ( x ) k C r − , which concludes the proof.3.5. Proof of Theorems 2.3 and 2.4.
Let us first prove Theorem 2.3. To thisend, fix any p > D the space of Fourier transforms of functions f ∈ L ∩ L p with power decay, namely, for which there exist constants A , a > f ( ξ ) ≤ A | ξ | − a for all | ξ | ≥
1. Then, since S ⊂ D ⊂ G , it is clear that D isdense in C ( R ) .Consider Φ : Σ → D ; in order to conclude, we show that | η x | (cid:0) − n − + ε , n − + ε (cid:1) decays as a power of n for all x ∈ Σ . Indeed, let us denote d η x = f x ( ξ ) d ξ , with UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 9 f x ∈ L ∩ L p , and let 0 < ˜ α = − ε < . Then, using Hölder inequality, with p + q = | η x | (cid:0) − n − ˜ α , n − ˜ α (cid:1) ≤ Z n − ˜ α − n − ˜ α | f x | ( ξ ) d ξ ≤ k f x k p (cid:13)(cid:13)(cid:13) ( − n − ˜ α , n − ˜ α ) (cid:13)(cid:13)(cid:13) q ≤ k f x k p ( n − ˜ α ) / q ≤ k f x k p ( n − ˜ α ) − / p . Therefore, Theorem 2.3 holds for any α of the form ˜ α ( − p ) = ( − ε )( − p ) ,with ε > f ∈ W is the Fourier transform of a function g ∈ L ∩ L , which satisfies k g k = k f k and k g k ≤ k f k W . This implies that any Lipschitz map Φ : Σ → W is Lipschitz also with respect to the total variation norm.Let us check that Φ satisfies the tightness condition (TC). Denote d η x ( ξ ) = f x d ξ , with f x ∈ L ∩ L . Again, it follows from [3, Theorem 4.2] that ξ f x ( ξ ) ∈ L and k ξ f x ( ξ ) k ≤ k Φ ( x ) ′ k . For any r ≥
2, by Cauchy-Schwartz and by Plancharelformula, we have | η x | ( R \ [ − r , r ]) = Z R \ [ − r , r ] | f x | ( ξ ) d ξ = Z R \ [ − r , r ] | f x | ( ξ ) + i | ξ | + i | ξ | d ξ ≤ (cid:13)(cid:13) R \ [ − r , r ] · ( + i | ξ | ) − (cid:13)(cid:13) · k ( + i | ξ | ) · | f x | ( ξ ) k ≤ ( π − − ( r )) k Φ ( x ) k W ≤ (cid:18) max x ∈ Σ k Φ ( x ) k W (cid:19) r − . The estimate | η x | (cid:0) − n − + ε , n − + ε (cid:1) = O (cid:0) n − + ε (cid:1) follows from Cauchy-Schwarz in-equality exactly as above. If in addition Φ has range in W ∩ L p , then the functions f x belong to L q , where p + q =
1, and one can conclude using Hölder inequalityagain. This finishes the proof.3.6.
Example.
We discuss a simple example, which shows that the bound in The-orem 2.4 cannot, in general, be improved. As local observable, let us consider anynon-negative ψ ( x , r ) = ψ ( r ) ∈ C ∞ c ( R ) which equals 1 in the interval (cid:2) − , (cid:3) and,as global observable, let Φ ( x , r ) = Φ ( r ) = + | r | . Then, Φ ∈ W ( R ) ∩ L p ( R ) for any p >
1, so that Theorem 2.4 implies that for any ε > C ≥ | cov ( Φ ◦ F n , ψ ) | = Z Σ × R ( Φ ◦ F n ) · ψ d ν ≤ Cn − + ε . Let us show that there is a lower bound of order exactly O ( n − ) .Lemma 3.1 implies that f is not cohomologous to zero. Moreover, by the CentralLimit Theorem, there exists a constant C ′ > n ∈ N sufficientlylarge, on a subset Y n ⊂ Σ of measure at least 1 /
2, the Birkhoff sums f n ( x ) = f ( x ) + · · · + f ( σ n − x ) are bounded by | f n ( x ) | ≤ C ′ √ n . In particular, for any x ∈ Y n and r ∈ (cid:2) − , (cid:3) , we have Φ ◦ F n ( x , r ) = Φ ( r + f n ( x )) = + r + | f n ( x ) | ≥ C ′ √ n . Thus, for any n ∈ N sufficiently large, it follows that Z Σ × R ( Φ ◦ F n ) · ψ d ν ≥ Z Y n × [ − , ]( Φ ◦ F n ) · ψ d ν ≥ ν (cid:18) Y n × (cid:20) − , (cid:21)(cid:19) C ′ √ n ≥ C ′ √ n . We have shown that there exists a constant C = ( C ′ ) − and, for any ε >
0, thereexists a constant C ε > Cn − ≤ cov ( Φ ◦ F n , ψ ) ≤ C ε n − + ε , hence the bound of Theorem 2.4 is, in this case, optimal.4. S KEW - PRODUCTS OVER ONE - SIDED SUBSHIFTS
To prove Theorem 3.8, we have to first prove analogous statements for one-sidedsubshifts. In this section, we discuss the case of skew-products over topologicallymixing one-sided subshifts of finite type.Let σ : X → X be a topologically mixing one-sided subshift of finite type, equippedwith a Gibbs measure µ = µ u with respect to the potential u . For 0 < θ <
1, thedistance d + θ and the space of Lipschitz functions F + θ are defined analogously tothe case of the two-sided shift. Let f + ∈ F + θ be a real-valued Lipschitz functionwith zero average, and consider the skew-shift(6) F + : X × R → X × R , F ( x , r ) = ( σ x , r + f + ( x )) . Denote by ν the infinite measure µ × Leb on X × R . For any pair of global andlocal observables Φ , ψ over X × R , define the analogous correlation functioncov ( Φ ◦ ( F + ) n , ψ ) : = Z X × R ( Φ ◦ ( F + ) n )( x , r ) · ψ ( x , r ) d ν ( x , r ) . Global and local observables for skew-shifts over one-sided subshifts.
Theclass of global and local observables we consider in this case are described below.In this setting, we require less regularity of the observables than in the case oftwo-sided shifts.
Definition 4.1 (Local observables – one-sided case) . Let L + ⊂ L ( ν ) be the spaceof functions ψ : X → S such that, for every ℓ ∈ N , the function x ∂ ℓ ψ ( x ) fromX to L ( R ) is Lipschitz. For every ψ ∈ L + , denote by Max ℓ ( ψ ) and Lip ℓ ( ψ ) theminimum constants such that (7) (cid:13)(cid:13) ∂ ℓ ψ ( x ) (cid:13)(cid:13) L ( R ) ≤ Max ℓ ( ψ ) and (cid:13)(cid:13) ∂ ℓ ψ ( x ) − ∂ ℓ ψ ( y ) (cid:13)(cid:13) L ( R ) ≤ Lip ℓ ( ψ ) d + θ ( x , y ) . Let us remark that, if ψ ∈ L + , then, for every fixed x ∈ X , the Fourier transform d ψ ( x ) of ψ ( x ) ∈ S is a Schwarz function as well. For any fixed ξ ∈ R , we denoteby b ψ ξ : X → C the function b ψ ξ ( x ) = d ψ ( x )( ξ ) . Lemma 4.2.
Let ψ ∈ L + . For every ξ ∈ R , we have b ψ ξ ∈ F + θ . Moreover, forevery ℓ ≥ , and for all ξ = we have (cid:13)(cid:13) b ψ ξ (cid:13)(cid:13) ∞ ≤ Max ℓ ( ψ ) ξ − ℓ and | b ψ ξ | θ ≤ Lip ℓ ( ψ ) ξ − ℓ . Proof.
For any ξ = x ∈ X , and ℓ ≥ | ξ ℓ d ψ ( x )( ξ ) | = | \ ∂ ℓ ψ ( x )( ξ ) | ≤ (cid:13)(cid:13)(cid:13) \ ∂ ℓ ψ ( x ) (cid:13)(cid:13)(cid:13) ∞ ≤ (cid:13)(cid:13) ∂ ℓ ψ ( x ) (cid:13)(cid:13) L ≤ Max ℓ ( ψ ) , hence sup x | b ψ ξ ( x ) | ≤ Max ℓ ( ψ ) ξ − ℓ . Similarly, for any x = y ∈ X , | ξ ℓ [ d ψ ( x )( ξ ) − d ψ ( y )( ξ )] | = | \ ∂ ℓ ψ ( x )( ξ ) − \ ∂ ℓ ψ ( y )( ξ ) | ≤ (cid:13)(cid:13) ∂ ℓ ψ ( x ) − ∂ ℓ ψ ( y ) (cid:13)(cid:13) L ≤ Lip ℓ ( ψ ) d + θ ( x , y ) , so that, for any fixed ξ =
0, we have | b ψ ξ | θ ≤ Lip ℓ ( ψ ) ξ − ℓ . (cid:3) Let us recall that A ⊂ L ∞ denotes the space of Fourier-Stieltjes transforms ofcomplex measures with finite total variation. UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 11
Definition 4.3 (Global observables – one-sided case) . Let G + ⊂ L ∞ ( ν ) be thespace of bounded functions Φ : X → A . For Φ ∈ G + , we define k Φ k G + : = sup x ∈ X k η x k TV , where, as usual, b η x = Φ ( x ) . In the next sections, we will deal only with the non-invertible case of the sew-product F + and we will often suppress the + in the notations introduced above,as it should not generate confusion. In Section 10 we will return to the invertiblesetting.4.2. Collapsed accessibility.
The property we need in the case of one-sided shiftswhich will replace the accessibility assumption is the following notion of collapsedaccessibility . Definition 4.4.
A Lipschitz function f : X → R has the collapsed accessibility property if there are constants C and N such that the following holds: for anyx ∈ X , t ∈ [ , ] , and n ≥ N, there is a sequence of pointsx , y , x , y , . . . y m , x m + such that (1) m ≤ N and x = x m + = x; (2) σ n x i = σ n y i ; (3) d ( y i , x i + ) ≤ Cr n ; and (4) t = ∑ mk = f n ( x k ) − f n ( y k ) . The adjective “collapsed” refers to the fact that local stable manifolds are col-lapsed to points when going from Σ × R to X × R . In order to prove Theorem 3.8, we will see in Section 10 that we can reduce anaccessible skew-product F to a skew-product F + over a one-sided shift such that f + enjoys the collapsed accessibility property.4.3. The one-sided version of the main theorem.
We state our main theoremin the case of skew-products over one-sided subshifts which have the collapsedaccessibility property. In Section 10, we will deduce Theorem 3.8 from Theorem4.5 below.
Theorem 4.5 (Quantitative global-local mixing for one-sided subshifts) . Assumethat f + , defined as in (6) , has the collapsed accessibility property. Then, for every ψ ∈ L + , for every Φ ∈ G + , for any k ∈ N , and for every ε > , there exists aconstant C = C ( Φ , ψ , k , ε ) > such that for every n ∈ N , | cov ( Φ ◦ F n , ψ ) | ≤ C (cid:16) LF ( Φ , n − + ε ) + n − k (cid:17) . The “low frequency” term LF ( Φ , · ) in Theorem 4.5 is defined exactly as in (5),except that the integral is on X instead of Σ .4.4. An expression for the correlation function.
The main tool to study the cor-relations is the transfer operator . We recall the relevant definitions.We denote by L = L σ : L ( µ ) → L ( µ ) the transfer operator for the base dynam-ics σ : X → X , namely the operator on L ( µ ) defined implicitly by Z X ( v ◦ σ ) w d µ = Z X v · ( Lw ) d µ , for v ∈ L ∞ ( µ ) and w ∈ L ( µ ) . Similarly, we denote by L F + : L ( ν ) → L ( ν ) thetransfer operator associated to F + , that is, the operator which, for every Φ ∈ L ∞ ( ν ) and ψ ∈ L ( ν ) , satisfies Z X × R ( Φ ◦ F + ) ψ d ν = Z X × R Φ · ( L F + ψ ) d ν . Explicitly, for any n ∈ N , we can write(8) ( L n w )( x ) = ∑ σ n y = x e u n ( y ) w ( y ) and ( L nF + ψ )( x , r ) = ∑ σ n y = x e u n ( y ) ψ ( y , r − f n ( y )) . For any z ∈ C , we let us further define the twisted transfer operator L z : L ( µ ) → L ( µ ) by ( L nz w )( x ) = ∑ σ n y = x e u n ( y ) − iz f n ( y ) w ( y ) , where u is the potential defining the Gibbs measure and u n its cocycle. Notice thatall the operators described above restrict to operators acting on F + θ . Proposition 4.6.
Let ψ ∈ L + and Φ ∈ G + . Then, for every n ∈ N we have Z X × R ( Φ ◦ ( F + ) n ) · ψ d ν = Z X Z ∞ − ∞ ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) . Proof.
By definition of the transfer operator L F + , we can write Z X × R Φ ◦ ( F + ) n ( x , r ) · ψ ( x , r ) d ν ( x , r ) = Z X × R Φ ( x , r ) · L nF + ψ ( x , r ) d ν = Z X Z ∞ − ∞ Φ ( x , r ) · L nF + ψ ( x , r ) d r d µ , where the applicability of the Fubini-Tonelli Theorem follows immediately fromthe definition of G + and L + . Since Φ ( x ) is the Fourier-Stieltjes transform of ameasure η x we get Z X × R ( Φ ◦ ( F + ) n ) · ψ d ν = Z X Z ∞ − ∞ (cid:18) Z ∞ − ∞ e − ir ξ d η x ( ξ ) (cid:19) L nF + ψ ( x , r ) d r d µ = Z X Z ∞ − ∞ Z ∞ − ∞ e ir ξ L nF + ψ ( x , r ) d η x ( ξ ) d r d µ . For every x ∈ X , we have Z ∞ − ∞ Z ∞ − ∞ | L nF + ψ ( x , r ) | d | η x | ( ξ ) d r ≤ k Φ k G + Z ∞ − ∞ | L nF + ψ ( x , r ) | d r ≤ k Φ k G + k ψ ( x ) k ≤ k Φ k G + Max ( ψ ) , thus we can again apply the Fubini-Tonelli Theorem to get Z X × R ( Φ ◦ ( F + ) n ) · ψ d ν = Z X Z ∞ − ∞ (cid:18) Z ∞ − ∞ e ir ξ L nF + ψ ( x , r ) d r (cid:19) d η x ( ξ ) d µ = Z X Z ∞ − ∞ \ L nF + ψ ( x , − ξ ) d η x ( ξ ) d µ = Z X Z ∞ − ∞ \ L nF + ψ ( x , ξ ) d η x ( ξ ) d µ . The conclusion follows by construction due to the equality \ L nF + ψ ( x , ξ ) = ( L n ξ ψ ξ )( x ) . (cid:3) UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 13
5. C
ANCELLATIONS FOR TWISTED TRANSFER OPERATORS
From Proposition 4.6, it is clear that, in order to estimate the correlations, weneed to study the twisted transfer operators L ξ , for real frequencies ξ ∈ R . The aimof this section is to show that the collapsed accessibility property can be exploitedto obtain some cancellations in the expression of L ξ .Let us fix a complex-valued Lipschitz function ˜ g : X → C , and let | ˜ g | = g . In thissection, we use a tilde to denote a complex-valued or “twisted” function, and thesame letter without a tilde to denote its absolute value. We denote by L : F + θ → F + θ the operator defined by ( L ˜ v )( x ) = ∑ σ y = x ˜ g ( y ) · ˜ v ( y ) , and by L : F + θ → F + θ the positive “untwisted” operator ( Lv )( x ) = ∑ σ y = x g ( y ) · v ( y ) . Up to conjugating L with a suitable multiplication operator, we can assume that L =
1, namely ∑ σ y = x g ( y ) = , for all x ∈ X .Notice that the for the operator L ξ defined in the previous section, we have˜ g = exp ( u + i ξ f ) , where u is the potential for the Gibbs measure µ .One can easily see that | L ˜ v ( x ) | ≤ Lv ( x ) . Moreover, recall that | · | θ is the Lips-chitz seminorm defined in (1). Then, the following Lasota-Yorke inequality holds,see [26, Proposition 2.1]. Lemma 5.1 (Basic inequality) . There exists a constant C > such that | L ˜ v | θ ≤ θ | ˜ v | θ + R k ˜ v k ∞ , where R = C | ˜ g | θ . By induction, ( L n v )( x ) = ∑ σ n y = x g n ( y ) v ( y ) and ( L n ˜ v )( x ) = ∑ σ n y = x ˜ g n ( y ) ˜ v ( y ) where g n and ˜ g n are the cocycles g n ( x ) = g ( x ) g ( σ ( x )) · · · g ( σ n − ( x )) and ˜ g n ( x ) = ˜ g ( x ) ˜ g ( σ ( x )) · · · ˜ g ( σ n − ( x )) . It follows that for all n ≥ | L n ˜ v | θ ≤ θ n | ˜ v | θ + R − θ k ˜ v k ∞ . Collapsed accessibility and cancellation pairs.
Let us fix a positive constant ε >
0, and an integer n ≥
1. We assume ε < and ε < − θ . Define H : = max (cid:26) , R − θ (cid:27) . A Lipschitz function ˜ v : X → C is a nice observable if | ˜ v | θ ≤ H and 1 − ε < v ( x ) < x ∈ X (as always, | ˜ v | = v ).We say L has ( ε , n ) -cancellation if for any observable ˜ v with | ˜ v | θ ≤ H and0 ≤ v ( x ) < x ∈ X , there is an integer 0 ≤ k ≤ n and a point x ∈ X suchthat | L k ˜ v ( x ) | ≤ − ε . We say L has strong ( ε , n ) -cancellation if for every nice observable ˜ v , there is a point x ∈ X such that | L k ˜ v ( x ) | ≤ − ε . One can see thatstrong ( ε , n ) -cancellation implies ( ε , n ) -cancellation.A pair of points ( x , y ) in X is a stable pair if σ n x = σ n y . We say a stable pair ( x , y ) is a cancellation pair for a nice observable ˜ v if | ˜ g n ( x ) ˜ v ( x ) + ˜ g n ( y ) ˜ g n ( y ) | ≤ g n ( x ) v ( x ) + g n ( y ) v ( y ) − ε . Lemma 5.2. If ( x , y ) is a cancellation pair for ˜ v, then | L n ˜ v ( p ) | ≤ − ε , where p = σ n x = σ n y.Proof. By definition of cancellation pair, we have | L n ˜ v ( p ) | ≤ | ˜ g n ( x ) ˜ v ( x ) + ˜ g n ( y ) ˜ g n ( y ) | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ σ n q = p , q = x , y ˜ g n ( q ) ˜ v ( q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ g n ( x ) v ( x ) + g n ( y ) v ( y ) − ε + ∑ σ n q = p , q = x , y g n ( q ) v ( q ) ≤ ∑ σ n q = p g n ( q ) ! − ε = − ε , where we used that L n = (cid:3) For a stable pair ( x , y ) define the phase of ( x , y ) asarg (cid:18) ˜ g n ( y ) ˜ g n ( x ) (cid:19) . Here, arg is the complex argument and so the phase is the angle between ˜ g n ( x ) and˜ g n ( y ) in the complex plane. For the most part, we can just think of this value as anangle. However, if we include it in an inequality, we will assume it is a real numberbetween − π and π .Define the stable tolerance of ( x , y ) as the number 0 < δ < π which satisfies1 − cos ( δ ) = ε (cid:18) g n ( x ) + g n ( y ) (cid:19) . Note that the right hand side must be less than two for this to be well defined. Inpractice, we will always choose ε small enough so that this is the case. Proposition 5.3.
Let ( x , y ) be a stable pair, and ˜ v a nice observable. If ( x , y ) is nota cancellation pair for ˜ v, then − δ ≤ arg (cid:18) ˜ g n ( x ) ˜ g n ( y ) ˜ v ( x ) ˜ v ( y ) (cid:19) ≤ δ , where δ is the stable tolerance of ( x , y ) . In other words, if s is the phase of ( x , y ) , then s + δ < arg (cid:18) ˜ v ( x ) ˜ v ( y ) (cid:19) < s − δ ignoring issues of the angle only being defined up to a multiple of 2 π .To prove the proposition, we first establish the following lemma. UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 15
Lemma 5.4.
Let z and z be non-zero complex numbers with α = arg ( z z ) . If ε (cid:18) | z | + | z | (cid:19) ≤ ( − cos ( α )) then | z + z | ≤ | z | + | z | − ε . Proof.
Write z = z + z and r k = | z k | for k = , ,
2. We wish to show that r ≤ ( r + r − ε ) . The cosine rule implies that r = r + r + r r cos ( α ) and so it is enough to show2 r r cos ( α ) ≤ r r − ε ( r + r ) + ε . This may be rewritten as ε (cid:18) r + r (cid:19) = ε r + r r r ≤ ( − cos ( α )) + ε r r . (cid:3) Proof of Proposition . We show the contrapositive. Define z = ˜ g n ( x ) ˜ v ( x ) and z = ˜ g n ( y ) ˜ v ( y ) . Then | z | = g n ( x ) v ( x ) ≥ ( − ε ) g n ( x ) , and a similar estimate holds for | z | . Using ε < and the definition of the stabletolerance, one sees that ε (cid:18) | z | + | z | (cid:19) ≤ ε − ε (cid:18) g n ( x ) + g n ( y ) (cid:19) ≤ ( − cos ( δ )) . Let α be the angle between z and z . If δ < α , then 1 − cos ( δ ) < − cos ( α ) andLemma 5.4 shows that ( x , y ) is a cancellation pair for ˜ v . (cid:3) For an arbitrary pair ( x , y ) of points in X , define the unstable tolerance as 0 ≤ δ < π such that sin ( δ ) = Hd ( x , y ) Note that x and y must be reasonably close for this to be well defined. Proposition 5.5. If ( x , y ) is a pair with unstable tolerance δ and ˜ v is a nice observ-able, then − δ ≤ arg (cid:18) ˜ v ( x ) ˜ v ( y ) (cid:19) ≤ δ . Again, we rely on a trigonometric lemma.
Lemma 5.6.
Let z and z be non-zero complex numbers with angle α = arg ( z z ) .If < α < π and − ε ≤ | z i | ≤ , then ( − ε ) sin ( α ) < | z − z | . Proof.
Assume | z | > | z | and consider the acute triangle defined by the points 0, z and z in complex plane. Split this triangle into two right triangles by addinga line segment from z to the opposite side of the triangle. This new segment haslength | z | sin ( α ) ≥ ( − ε ) sin ( α ) and so the line segment from z to z has lengthat least ( − ε ) sin ( α ) . (cid:3) Proof of Proposition . Let z = ˜ v ( x ) and z = ˜ v ( y ) and let α be the angle be-tween them. The above lemma and the definition of “nice” together imply that ( − ε ) sin ( α ) ≤ | z − z | ≤ Hd ( x , y ) . Since ε < by assumption, the result follows. (cid:3) A us-cycle is a (finite) sequence of points in X : x , y , x , y , . . . , y m , x m + where x = x m + and each pair ( x k , y k ) is a stable pair. The tolerance of the cycleis the sum of the stable tolerances of the pairs ( x , y ) , ( x , y ) , . . . , ( x m , y m ) and the unstable tolerances of the pairs ( y , x ) , ( y , x ) , . . . , ( y m , x m + ) . We only consider us-cycles for which this tolerance is well defined. The phase ofthe cycle is arg (cid:18) ˜ g ( y ) ˜ g ( x ) ˜ g ( y ) ˜ g ( x ) · · · ˜ g ( y m ) ˜ g ( x m ) (cid:19) . That is, the phase of the cycle is the sum of the phases of the individual stable pairs(up to a multiple of 2 π ). As defined, the phase is a number in ( − π , π ] . We willonly consider cycles where the phase is positive. Proposition 5.7.
If there is a us-cycle where the phase is greater than the tolerance,then L has strong ( ε , n ) -cancellation.Proof. Let ˜ v be a nice observable. Our goal is to show that one of the stable pairsin the cycle is a cancelling pair for ˜ v . We assume none of them is a cancellingpair and derive a contradiction. Let S be the phase of the cycle and δ = δ s + δ u bethe tolerance, where δ s is the sum of the stable tolerances and δ u is the sum of theunstable tolerances. We are assuming 0 < δ < S . Proposition 5.3 implies that S − δ s < arg (cid:18) ˜ v ( x ) ˜ v ( y ) ˜ v ( x ) ˜ v ( y ) · · · ˜ v ( x m ) ˜ v ( y m ) (cid:19) < S + δ s and Proposition 5.5 implies that − δ u < arg (cid:18) ˜ v ( x ) ˜ v ( y ) ˜ v ( x ) ˜ v ( y ) · · · ˜ v ( x m + ) ˜ v ( y m ) (cid:19) < δ u . Since x = x m + , the complicated product in the middle of each inequality is actu-ally the same complex number and so we get S − δ s < δ u , a contradiction. (cid:3) Cancellation by frequency.
We now apply the results above to the specificcase of the operators L ξ defined in the previous section, namely to the case ( L ξ ˜ v )( x ) = ∑ σ y = x ˜ g ξ ( y ) · ˜ v ( x ) , where ˜ g ξ = exp ( u + i ξ f ) . To simplify the presentation we only consider positive ξ , but analogous results will hold for negative frequencies.One can show that the Lipschitz norm of ˜ g ξ satisfies | ˜ g ξ | θ ≤ | g | θ + ξ | f | θ , and soeach twisted operator satisfies a Lasota-Yorke inequality | L ξ ˜ v | θ ≤ θ | ˜ v | θ + R ξ k ˜ v k ∞ ,where R ξ grows linearly in ξ , see Lemma 5.1.The notion of a “nice observable” will also depend on the frequency. In par-ticular, the value H from the previous section depends on ξ and so we denote it UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 17 by H ξ = − θ R ξ , which also grows linearly in ξ . Define a constant G = inf { g ( x ) : x ∈ X } and an exponent α > θ α G =
1. Note that G and α areindependent of the frequency.We now show that accessibility of the skew product leads to cancellation of thesetwisted operators and give quantitative estimates of the amount of cancellation. Proposition 5.8.
Suppose f + has the collapsed accessibilty property and ξ > is given.Then there are positive constants A and B such that if ξ ≥ ξ and ε ξ = AG n ξ where n ξ is the smallest integer which satisfies θ n ξ < B ξ , then L ξ has strong ( ε ξ , n ξ ) -cancellation. Remark 5.9.
One can see from the definitions of ε ξ and n ξ that ε ξ = A θ α n ξ ≥ θ AB α ξ − α .Proof. Assume without loss of generality that 0 < ξ < π . The overall strategy ofthe proof is to use collapsed accessibility to show that, for any frequency ξ ≥ ξ ,there is a us-cycle with phase equal to ξ and tolerance less than ξ . Proposition5.7 then gives cancellation.Let C and N be as given as in Definition 4.4 of collapsed accessibility . Thenthere is a constant 0 < a < < δ < π which satisfies either1 − cos ( δ ) ≤ a or sin ( δ ) ≤ a also satisfies δ < N ξ . Since H ξ grows linearly in ξ , there is B > CH ξ ≤ aB ξ , for all ξ ≥ ξ . Up to increasing the valueof B , we can also ensure that n > N for any integer n which satisfies θ n < B ξ .Define A = a .Now consider a specific frequency ξ ≥ ξ and use n = n ξ and ε = ε ξ definedas in the statement of the proposition. Using this n and t = ξξ , there is a sequenceof points x , y , x , y , . . . , y m , x m + satisfying Definition 4.4. This sequence is a us-cycle for L ξ and has phase equal to ξ . If δ is the stable tolerance of a pair ( x k , y k ) , then 1 − cos ( δ ) = ε (cid:18) g n ( x k ) + g n ( y k ) (cid:19) ≤ ε G n = a . If instead δ is the unstable tolerance of a pair ( y k , x k + ) , thensin ( δ ) = H ξ d ( y k , x k + ) ≤ CH ξ θ n ≤ aB ξθ n ≤ a . Together, these estimates show that the total tolerance of the us-cycle is less than ξ and so Proposition 5.7 gives cancellation. (cid:3)
6. C
ONTRACTION
In this section, we show how to obtain some estimates on the norm of the opera-tor L ξ . For high frequencies, we exploit the cancellations obtained in the previoussection, while, for low frequencies, we apply some standard results from the per-turbation theory of bounded linear operators.6.1. High frequencies.
Recall that we defined H : = max (cid:8) , R − θ (cid:9) . It will be con-venient to define the following norm on F + θ : let k ˜ v k H : = max (cid:26) k ˜ v k ∞ , | ˜ v | θ H (cid:27) . Notice that the norms k · k H and k · k θ are equivalent, namely k ˜ v k H ≤ k ˜ v k θ ≤ H k ˜ v k H . In this section, we will prove the following result.
Proposition 6.1.
Suppose that f + has the collapsed accessibility property, and let ξ > be given. Then, there exists positive constants A , B > and an exponent β > such that for all ξ ≥ ξ we have k L N ξ k H ≤ − A ξ − β , for all N ≥ B | log ξ | . We start by proving some simple preliminary results.
Lemma 6.2.
For any given ξ > , if ˜ v ∈ F + θ , then k L ξ ˜ v k H ≤ k ˜ v k H .Proof. Clearly, k L ξ ˜ v k ∞ ≤ k ˜ v k ∞ ≤ k ˜ v k H . From the Basic Inequality in Lemma 5.1we also get | L ξ ˜ v | θ H ≤ θ | ˜ v | θ H + RH k ˜ v k ∞ ≤ (cid:18) θ + RH (cid:19) k ˜ v k H ≤ k ˜ v k H , since H > R / ( − θ ) . This completes the proof (cid:3) Let us recall that, from the definition of Gibbs measure, it follows that there existconstants C u , d such that for any ball B ( x , r ) centered at x ∈ X with radius r ≥ µ ( B ( x , r )) ≥ C u r d . We will also use the fact that the untwisted transfer operator L on F + θ has aspectral gap, namely the following well-known result, see, e.g., [26, Theorem 2.2]. Lemma 6.3.
There exist a bounded operator N : F + θ → F + θ , a real number < δ < , and a constant C > such that for all n ∈ N we have k N k θ ≤ C δ n , and forall ˜ v ∈ F + θ , L n ( ˜ v ) = Z X ˜ v d µ + N n ( ˜ v ) . We have the following result.
Lemma 6.4.
There exists a constant C > such that the following holds. For any ε > , ℓ ≥ , and ˜ v ∈ F + θ with | ˜ v | θ ≤ ℓ , if | ˜ v ( ¯ x ) | ≤ − ε for a point ¯ x ∈ X , then Z X v d µ ≤ − C (cid:16) ε ℓ (cid:17) d ε . Proof.
Define w = − v and note that w ( ¯ x ) ≥ ε and the Lipschitz semi-norm of w satisfies | w | θ = | v | θ ≤ | ˜ v | θ ≤ ℓ . If B is the ball centered at ¯ x of radius r = ε ℓ , then w ( x ) ≥ ε for all x ∈ B , so Z X w d µ ≥ ε µ ( B ) ≥ ε C u (cid:16) ε ℓ (cid:17) d . Since R X v d µ = − R X w d µ , the result follows. (cid:3) Lemma 6.5.
There exist constants ¯ A , ¯ B > such that the following holds. Assumethat L = L ξ has ( ε , n ) -cancellation. Then, for every ˜ v ∈ F + θ with k ˜ v k H ≤ , andfor any N ≥ N : = ⌊− ¯ B log ( ε / H ) ⌋ , we have k L N + n ˜ v k H ≤ − ¯ A ε d + H d . UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 19
Proof.
Let N and N be the minimum integers which satisfy3 CH δ N ≤ C ε d + H d and 2 θ N ≤ C ε d + H d , and let N = N + N . It is clear from the definition that there exists a constant¯ B > N = ⌊− ¯ B log ( ε / H ) ⌋ .For every x ∈ X , | L N + n ˜ v ( x ) | ≤ | L N ( | L n ˜ v | )( x ) | ≤ Z X | L n ˜ v | d µ + | N N ( | L n ˜ v | )( x ) | , where, by Lemma 6.3 and (9), | N N ( | L n ˜ v | )( x ) | ≤ C δ N k L n ˜ v k θ ≤ C δ N ( + θ n | ˜ v | θ + H ) ≤ CH δ N ≤ C ε d + H d , where the last inequality follows from the definition of N . From Lemma 6.2, wehave | L n ˜ v | θ ≤ H k ˜ v k H ≤ H . Thus, Lemma 6.4 applied to | L n ˜ v | gives us | L N + n ˜ v ( x ) | ≤ − C (cid:16) ε H (cid:17) d ε + C ε d + H d . Therefore, we obtain(11) k L N + n ˜ v k ∞ ≤ k L N + n ˜ v k ∞ ≤ − C ε d + H d . Finally, the inequality (9) gives us | L N + n ˜ v | θ ≤ θ N | L N − N + n ˜ v | θ + H k L N − N + n ˜ v k ∞ ≤ H (cid:0) θ N + k L N + n ˜ v k ∞ (cid:1) . By the definition of N and (11), we conclude | L N + n ˜ v | θ ≤ H (cid:0) θ N + k L N + n ˜ v k ∞ (cid:1) ≤ H (cid:18) − C ε d + H d (cid:19) . The inequality above and (11) conclude the proof. (cid:3)
We are in position to complete the proof of Proposition 6.1
Proof of Proposition . By Proposition 5.8, L ξ has ( ε ξ , n ξ ) -cancellations, with ε ≥ A ξ − α and n ξ ≤ B | log ξ | , for some positive constants A , B . Therefore, byLemma 6.5, for every N ≥ N + n ξ we have k L N ξ k H ≤ − ¯ A ε d + H d ≤ − A ξ − ( α d + α + d ) , for some constant A >
0. By the definitions of N and n ξ , there exists a constant B > N + n ξ ≤ B | log ξ | . (cid:3) Low frequencies.
We now want to estimate the norm of L ξ for small ξ ∈ R .Let us notice that there exists ξ > ≤ ξ ≤ ξ we have H = max (cid:8) , R − θ (cid:9) =
1, so that k · k H ≤ k · k θ ≤ k · k H . We will prove the followingbound. Proposition 6.6.
There exist κ > and a constant A κ > such that, for all < ξ < κ and for all n ≥ , we have k L n ξ k H ≤ ( − A κ ξ ) n . Let us recall that the family of operators z L z is analytic for z ∈ C . Thisensures that we can apply classical results from analytic perturbation theory tostudy the spectrum of bounded linear operators, see in particular [22, TheoremVII.1.8]. In our case, since the operator L = L has a spectral gap, we can deducethe following result, see [26, Chapter 4], [19, Proposition 2.3] and [31, p.15]. Theorem 6.7.
There exists a κ > such that the twisted transfer operator L z on F + θ has a spectral gap for all | z | < κ . Moreover, there exist λ z ∈ C and linearoperators P z and N z such that L z = λ z P z + N z and which satisfy the followingproperties: (1) λ z , P z and N z are analytic on the disk {| z | < κ } , (2) P z is a projection and its range has dimension 1, (3) P z N z = N z P z = , (4) the spectral radius ρ ( N z ) of N z satisfies ρ ( N z ) < λ z − δ , for some δ inde-pendent of z. In our case, we restrict to real frequencies 0 < ξ < κ . For the proof of thefollowing lemma, see [26, Chapter 4] and [31, Section 4]. Lemma 6.8.
With the notation of Theorem , there exist constants A κ , B κ > such that for all < ξ < κ we have (cid:12)(cid:12) λ ξ − ( − A κ ξ ) (cid:12)(cid:12) ≤ B κ ξ . The fact that A κ is strictly positive follows from the fact that f + is not cohomol-ogous to zero, see Lemma 3.1. Proof of Proposition . By Theorem 6.7, up to choosing a smaller κ , we canassume that ρ ( N ξ ) < | λ ξ | ≤ − A κ ξ for all 0 < ξ ≤ κ . Therefore, for any ˜ v ∈ F + θ , we have k L n ξ ˜ v k H ≤ k L n ξ ˜ v k θ ≤ ( | λ ξ | n + ρ ( N ξ ) n ) k ˜ v k θ ≤ ( − A κ ξ ) n k ˜ v k H , which proves the result. (cid:3)
7. R
APID DECAY
In Section 8, we will use the contraction results established for the twisted trans-fer operator L ξ in the previous section in order to prove rapid mixing. In thissection, we give several technical propositions in an abstract setting which encap-sulate most of the difficult inequalities involved in the proof. Definition 7.1.
Consider a function w : A ⊆ ( , ∞ ) → R . We say w ( ξ ) decaysrapidly in ξ if for each ℓ ≥ , there is a constant C such that | w ( ξ ) | ≤ C ξ − ℓ for all ξ . We say a sequence { s n } decays rapidly in n if for each ℓ ≥ , there is a constantC such that | s n | ≤ Cn − ℓ for all n. Proposition 7.2.
Suppose A, B, β , and ξ are positive constants and that { w n } is adecreasing sequence of non-negative functions of the form w n : [ ξ , ∞ ) → [ , ] . Ifw ( ξ ) decays rapidly in ξ andw n + N ( ξ ) ≤ ( − A ξ − β ) w n ( ξ ) for all N > B log ( ξ ) , then the sequence { s n } defined by s n = sup ξ w n ( ξ ) decays rapidly in n. In order to prove this, we first give a lemma which establishes for each fixed ξ an exponential rate of decay of the sequence { w n ( ξ ) } . UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 21
Lemma 7.3.
In the setting of Proposition , there are constants D and γ suchthat w n + K ( ξ ) < e w n ( ξ ) for all K > D ξ γ .Proof. Consider a specific ξ and let k and N be the smallest integers such that k > A ξ β and N > B log ( ξ ) . Then ( − A ξ − β ) k < exp ( − kA ξ − β ) < exp ( − ) and so w n + kN ( ξ ) ≤ ( − A ξ − β ) k w n ( ξ ) ≤ e w n ( ξ ) . If we choose an exponent γ > β , then there is a constant D such that kN ≤ ( A ξ β + )( B log ξ + ) ≤ D ξ γ . Moreover, this constant D may be chosen uniformly for all ξ . If K > D ξ γ , then K > kN and w n + K ( ξ ) ≤ w n + kN ( ξ ) . (cid:3) Proof of Proposition . We will show for any q > Q suchthat s n < ε for all 0 < ε < n > Q ε − q . One can see that this condition impliesthat { s n } decays rapidly. Let D and γ be as in the above lemma and choose aninteger ℓ > γ / q . As w ( ξ ) decays rapidly in ξ , there is C such that w ( ξ ) < C ξ − ℓ for all ξ in the domain. For a given ε > • let a > Ca − ℓ = ε , • let j be the smallest integer such that e − j < ε , and • let K be the smallest integer such that K > Da γ .Now consider a frequency ξ . If ξ > a , then w jK ( ξ ) ≤ w ( ξ ) < ε . If instead ξ ≤ a ,then K > D ξ γ which implies w jK ( ξ ) ≤ e − j w ( ξ ) < ε . Together, these imply that s n < ε for all n > jK . Since a γ = C ε − γ /ℓ and q > γ /ℓ , there is a constant Q suchthat jK ≤ ( log ( ε − ) + )( Da γ + ) < Q ε − q holds uniformly for all ε . (cid:3) Proposition 7.4.
Suppose A, and ξ are positive constants, < α < , and { w n } is a decreasing sequence of non-negative functions of the form w n : ( , ξ ] → [ , ] .If w n + k ( ξ ) ≤ ( − A ξ ) k w n ( ξ ) for all ξ , k and n, then the sequence { s n } definedby s n = sup { w n ( ξ ) : n − α ≤ ξ ≤ ξ } decays rapidly in n.Proof. If n − α ≤ ξ , then w n ( ξ ) ≤ ( − A ξ ) n ≤ ( − nA ξ ) ≤ ( − n − α A ) ,and, since 1 − α >
0, one can show that 4 exp ( − n − α A ) decays rapidly in n . (cid:3) Propositions 7.2 and 7.4 are enough to establish rapid mixing in the setting ofskew products over one-sided shifts. However, to handle two-sided shifts, we willneed the following more technical results.
Proposition 7.5.
Let A, B, β , ξ and θ be positive constants. Let v n , m : [ ξ , ∞ ) → [ , ∞ ) be a collection of functions defined for all integers n ≥ and m ≥ and letw : [ ξ , ∞ ) → [ , ∞ ) be a bounded function. Suppose for all n , m ≥ and ξ ≥ ξ that (1) v n , m ( ξ ) ≤ v n + , m ( ξ ) , (2) v n + N , m ( ξ ) ≤ ( − A ξ − β ) v n , m ( ξ ) when N > B log ( ξ ) , (3) v , m ( ξ ) ≤ θ − m w ( ξ ) , and (4) w ( ξ ) decays rapidly in ξ . Then, for any c > , the sequence { t n } defined byt n = sup { v n , m ( ξ ) : ξ > ξ and m < c log ( n ) } decays rapidly in n.Proof. As w ( ξ ) is bounded, we may without loss of generality assume that w ( ξ ) ≤ ξ . Define w n ( ξ ) = sup m θ m v n , m ( ξ ) . One can verify that { w n } satisfies thehypotheses of Proposition 7.2. Hence sup ξ w n ( ξ ) decays rapidly in n , meaningthat for a given ℓ , there is C such that v n , m ( ξ ) < C θ − m n − ℓ for all m and n . If m < c log ( n ) , then θ − m < θ − c log ( n ) = n − c log ( θ ) ⇒ v n , m ( ξ ) < Cn − ℓ − c log ( θ ) . From this, one can see that { t n } decays rapidly in n . (cid:3) Proposition 7.6.
Let A, β , ξ and θ be positive constants, and < α < . Letv n , m : ( , ξ ] → [ , ∞ ) be a collection of functions defined for all integers n ≥ andm ≥ , and let w : ( , ξ ] → [ , ∞ ) be a bounded function. Suppose for all n , m , k ≥ and ξ ≤ ξ that (1) v n , m ( ξ ) ≤ v n + , m ( ξ ) , (2) v n + k , m ( ξ ) ≤ ( − A ξ ) k v n , m ( ξ ) , (3) v , m ( ξ ) ≤ θ − m w ( ξ ) .Then for any c > , the sequence { t n } defined byt n = sup { v n , m ( ξ ) : n − α < ξ ≤ ξ and m < c log ( n ) } decays rapidly in n.Proof. This follows from Proposition 7.4 using the proof of Proposition 7.5. (cid:3)
8. P
ROOF OF T HEOREM ψ ∈ L + and Φ ∈ G + be given, and fix k ∈ N and 0 < α < /
2. Recall thatthe global observable Φ defines a complex measure η x for each x ∈ X and that thereis a uniform constant M = k Φ k G + such that k η x k TV ≤ M for all x ∈ X . The Fouriertransform of the local observable ψ is a function of the form b ψ : X × R → C where,for each frequency ξ , the function b ψ ξ : X → C defined by b ψ ξ ( x ) = d ψ ( x )( ξ ) isHölder and lies in F + θ .By Proposition 4.6, we havecov ( Φ ◦ F n , ψ ) = Z X Z ∞ − ∞ ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) − ν av ( Φ ) ν ( ψ ) . We will estimate the correlations by splitting the frequencies ξ ∈ R into the cases ξ =
0, 0 < | ξ | < n − α , and | ξ | > n − α . In fact, we only consider ξ ≥ ξ < Lemma 8.1.
There exist constants C > and < δ < such that (cid:12)(cid:12)(cid:12)(cid:12) Z X Z { } ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) − ν av ( Φ ) ν ( ψ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ CM ( Max ( ψ ) + Lip ( ψ )) δ n , for all n. UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 23
Proof.
Recalling that L = L is the transfer operator associated to σ , we have Z { } ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) = η x ( { } )( L n b ψ )( x ) . By Lemma 6.3, there exists a constant C > < δ < (cid:12)(cid:12)(cid:12)(cid:12) ( L n b ψ )( x ) − Z X b ψ ( x ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = | ( L n b ψ )( x ) − ν ( ψ ) | ≤ C δ n k b ψ k θ , where we used that b ψ ( x ) = R R ψ ( x , r ) d r . Hence, by Lemma 3.5 and Lemma 4.2,we conclude (cid:12)(cid:12)(cid:12)(cid:12) Z X Z { } ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) − ν av ( Φ ) ν ( ψ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) Z X | η x | ( { } ) d µ ( x ) (cid:19) k b ψ k θ δ n ≤ CM ( Max ( ψ ) + Lip ( ψ )) δ n . (cid:3) Lemma 8.2.
We have that (cid:12)(cid:12)(cid:12)(cid:12) Z X Z ( , n − α ) ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Max ( ψ ) LF ( Φ , n − α ) , for all n.Proof. Since, by Lemma 4.2, we have k L n ξ b ψ ξ k ∞ ≤ k b ψ ξ k ∞ ≤ Max ( ψ ) , we obtain (cid:12)(cid:12)(cid:12)(cid:12) Z X Z ( , n − α ) ( L n ξ b ψ ξ )( x ) d η x ( ξ ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Max ( ψ ) Z X | η x | (cid:16) ( , n − α ) (cid:17) d µ ( x ) ≤ Max ( ψ ) LF ( Φ , n − α ) , which settles the proof. (cid:3) Lemma 8.3.
The sequence (cid:26) Z X Z [ n − α , ∞ ) L n ξ b ψ ξ d η x ( ξ ) d µ ( x ) (cid:27) n ≥ decays rapidly in n.Proof. For each n , define a function w n : [ , ∞ ) → [ , ∞ ) by w n ( ξ ) = k L n ξ b ψ ξ k H . By Lemma 6.2, w n is a decreasing sequence of functions, and by Lemma 4.2, w ( ξ ) is a bounded function which (in the notation of Section 7) decays rapidly in ξ . Upto rescaling ψ , we may freely assume that w takes values in [ , ] .Proposition 6.6 implies that w n restricted to ( , κ ] satisfies the hypotheses ofProposition 7.4. We then fix ξ = κ , so that Proposition 6.1 implies that w n re-stricted to [ ξ , ∞ ) satisfies the hypotheses of Proposition 7.2. Hence, the sequencedefined by s n = sup { w n ( ξ ) : n − α ≤ ξ < ∞ } decays rapidly in n . Note that k L n ξ b ψ ξ k ∞ ≤ k L n ξ b ψ ξ k H and so, for each n , (cid:12)(cid:12)(cid:12)(cid:12) Z [ n − α , ∞ ) L n ξ b ψ ξ d η x ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k η x k TV s n As µ is a probability measure, it follows that (cid:12)(cid:12)(cid:12)(cid:12) Z X Z [ n − α , ∞ ) L n ξ b ψ ξ d η x ( ξ ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ms n , where M is the uniform bound on k η x k TV . (cid:3)
9. F
ROM ACCESSIBILITY TO COLLAPSED ACCESSIBILITY
In this section, we relate the notion of accessibility for a skew-product F as in(2) to the property of collapsed accessibility defined in Section 4.For a two sided shift σ : Σ → Σ , let X be the corresponding one sided shift andlet π : Σ → X be the projection. Note that π is a continuous, surjective, open map.We also write x + for π ( x ) . For x ∈ Σ , define W s ( x ) = π − π ( x ) . In other words, y ∈ W s ( x ) if and only if x + = y + . For n ∈ Z , define W sn ( x ) = σ − n W s ( σ n x ) and note that W s ( x ) ⊂ W s ( x ) ⊂ W s ( x ) ⊂ · · · is an increasing sequence whose union is W s ( x ) . For a subset U ⊂ Σ , write W sn ( U ) = [ x ∈ U W sn ( U ) . Lemma 9.1.
If U ⊂ Σ is open, then W sn ( U ) is open for all n. If K ⊂ Σ is compact,then W sn ( K ) is compact for all n.Proof. Since π is an open map, W s ( U ) = π − π ( U ) is an open set. Since σ n is adiffeomorphism W sn ( U ) = σ − ( W s ( σ n U )) is also open. A similar proof holds forcompact sets. (cid:3) Lemma 9.2.
For points x and y in Σ and n ∈ Z , the following are equivalent: (1) y ∈ W sn ( x ) , (2) dist ( σ n + k x , σ n + k y ) ≤ r k for all k ≥ .Proof. One can show that each of these conditions is equivalent to the sequencesof symbols for x and y satisfying x m = y m for all m ≥ n . (cid:3) Instead of projecting onto the future x x + , we can analogously project ontothe past x x − . Define local unstable manifolds by y ∈ W u ( x ) if and only if x − = y − , and for n ∈ Z define W un ( x ) = σ n W s ( σ − n x ) . Analogous versions of theabove lemmas hold for these manifolds.Let us now consider the skew-product (2). Writing p = ( x , s ) and q = ( y , t ) , wedefine local stable manifolds by p ∈ W sn ( q ) if and only if p ∈ W s ( q ) and x ∈ W sn ( y ) .Define local unstable manifolds analogously. For points p and q and an integer n >
0, a us-N-path from p to q is a sequence p = p , p , . . . p n = q such that n ≤ N and for each 0 ≤ k < n either p k + ∈ W sN ( p k ) or p k + ∈ W uN ( p k ) .For a point p , define AC N ( x ) by q ∈ AC N ( x ) if and only if there is a us - N -pathfrom p to q . Note that AC N ( x ) form an increasing sequence whose union is AC ( p ) .For a subset U ⊂ Σ × R , define AC N ( U ) = [ x ∈ U AC N ( U ) . Lemma 9.3.
If U ⊂ Σ × R is open, then AC N ( U ) is open for all n ≥ . If K ⊂ Σ × R is compact, then AC N ( K ) is compact for all n ≥ .Proof. This follows directly from Lemma 9.1. (cid:3)
Proposition 9.4.
Let K be a compact subset of Σ × R such that int ( K ) = K. Ifp ∈ Σ × R is such that K ⊂ AC ( p ) , then there is N such that K ⊂ AC N ( p ) . UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 25
Proof.
Since AC N ( p ) is an increasing sequence of compact sets and K is a Bairespace, there is N such that AC N ( p ) contains a non-empty open subset U ⊂ K .Since AC N ( U ) is an increasing sequence of open sets whose union contains thecompact set K , there is N such that K ⊂ AC N ( U ) . Then K ⊂ AC N + N ( p ) . (cid:3) We have the following result.
Proposition 9.5.
Let f : X → R be a Lipschitz function. If the skew productF : Σ × R → Σ × R , ( x , t ) ( σ x , t + f ( x + )) is accessible, then f has the collapsed accessibility property.Proof. Let K = Σ × [ , ] . By Proposition 9.4, there is a uniform constant N suchthat AC N ( p ) contains K for any p ∈ K . With N fixed, let x ∈ X , t ∈ [ , ] , and n ≥ p , q , p , q , . . . , q m , p m + such that(1) m ≤ N , p = F N − n ( x , ) , and p m + = F N − n ( x , t ) ;(2) p k ∈ W sN ( q k ) ; and(3) p k + ∈ W uN ( q k ) .Applying F N − n to this sequence, we define ( a k , s k ) = F N − n ( p k ) and ( b k , t k ) = F N − n ( q k ) which satisfy b k ∈ W sn ( a k ) and a k + ∈ W u N − n ( b k ) . As ( a k , s k ) and ( b k , t k ) are on the same stable manifold in Σ × R , it follows that t k − s k = f n ( b + k ) − f n ( a + k ) . By the unstable analogue of Lemma 9.2, one can show that d ( b k , a k + ) ≤ r n − n + .That is, d ( b k , a k + ) ≤ Cr n where C = r − N . One can then check that x k = a + k and y k = b + k satisfy all of the conditions in the definition of collapsed accessibility. (cid:3)
10. P
ROOF OF T HEOREM
Step 1: f only depends on future coordinates. Let us start with a prelimi-nary step: we show that we can assume that the function f in (2) only depends onthe future coordinates. From [26, Proposition 1.2], we inherit the following result. Lemma 10.1.
There exist h ∈ F √ θ and f + ∈ F + √ θ such that f = f + + h − h ◦ σ . When reducing to a one-sided shift, we will encounter some loss in regularity asin the previous lemma: the functions h and f + are Holder with exponent 1 /
2. Wecan however replace θ with √ θ in the definition of the distance d θ to make themLipschitz. We remark that this is not an issue, and we will freely replace θ with asuitable choice that makes the functions Lipschitz.For any Φ ∈ G and ψ ∈ L , using Lemma 10.1, we can writecov ( Φ ◦ F n , ψ ) = Z Σ Z ∞ − ∞ Φ ( σ n x , r + f n ( x )) · ψ ( x , r ) d r d µ ( x )= Z Σ Z ∞ − ∞ Φ ( σ n x , r + f + n ( x ) + h ( x ) − h ( σ n x )) · ψ ( x , r ) d r d µ ( x ) . Let us define Φ h ( x , r ) = Φ ( x , r − h ( x )) and ψ h ( x , r ) = ψ ( x , r − h ( x )) . We changevariable s = r + h ( x ) and we getcov ( Φ ◦ F n , ψ ) = Z Σ Z ∞ − ∞ Φ ( σ n x , s + f + n ( x ) − h ( σ n x )) · ψ ( x , s − h ( x )) d s d µ ( x )= Z Σ Z ∞ − ∞ ( Φ h ◦ F n )( x , r ) · ψ h ( x , r ) d r d µ ( x ) , where the skew-product F is defined by F ( x , r ) = ( σ x , r + f + ( x )) . The map H ( x , r ) = ( x , r + h ( x )) used in the change of variable above is a conjugacy between F and F , namely H ◦ F = F ◦ H . Moreover, H is uniformly continuous (moreprecisely, it is Lipschitz with respect to the distance d √ θ , exactly as h ), hence itpreserves stable and unstable manifolds. In particular, F is accessible.The initial claim follows from the following lemma, whose proof is containedin the Appendix C. Lemma 10.2.
With the notation above, ψ h ∈ L with ν ( ψ h ) = ν ( ψ ) , and Φ h ∈ G with ν av ( Φ h ) = ν av ( Φ ) . Moreover, for every x ∈ Σ , we have | ( η h ) x | = | η x | , where [ ( η h ) x = Φ h ( x ) and b η x = Φ ( x ) . Step 2: observables only depend on future coordinates.
In the previoussubsection, we have seen that we can assume that f = f + ∈ F + θ (up to replacing θ with √ θ ). We now show that we can replace the observables Φ = Φ h ∈ G and ψ = ψ h ∈ L with observables in G + and in L + respectively: this is the contentof Proposition 10.3 below. The proof follows the same lines as in [13]; howeverin our case there are some additional difficulties in showing that the functions de-fined belong to G + and L + . In particular, we will need to use the assumption(TC) to ensure some compactness property in A . We postpone the proof to theAppendix C. Proposition 10.3.
Let Φ ∈ G and ψ ∈ L . There exist constants K , M ( Φ ) ≥ ,sequences { Φ m } m ∈ N ⊂ G + , { ψ } m ∈ N ⊂ L + , and, for every ℓ ∈ N , there exist con-stants M ( ψ , ℓ ) and L ( ψ , ℓ ) such that the following properties hold for all ℓ, m , n ∈ N and x ∈ X : (i) ν av ( Φ m ) = ν av ( Φ ) and ν ( ψ m ) = ν ( ψ ) , (ii) k Φ ◦ F m ( x , · ) − Φ m ( x , · ) k ≤ M ( Φ ) θ m , and k Φ ( x ) k ≤ M ( Φ ) , (iii) Max ℓ ( ψ m ) ≤ M ( ψ , ℓ ) and Lip ℓ ( ψ m ) ≤ θ − m L ( ψ , ℓ ) , (iv) | cov ( Φ ◦ F n , ψ ) − cov ( Φ m ◦ ( F + ) n , ψ m ) | ≤ K θ m . From Proposition 9.5, it follows that the function f + in the definition of theone-sided skew-product F + has the collapsed accessibility property.10.3. Step 3: end of the proof.
We are now ready to prove Theorem 3.8. Let Φ ∈ G and ψ ∈ L , and fix k ∈ N and 0 < α < /
2. Consider the sequence offunctions { ψ } m ∈ N ⊂ L + given by Proposition 10.3. By Lemma 4.2, their Fouriertransforms ( c ψ m ) ξ satisfy k ( c ψ m ) ξ k ∞ ≤ M ( ψ , ℓ ) ξ − ℓ and | ( c ψ m ) ξ | θ ≤ θ − m L ( ψ , ℓ ) ξ − ℓ . If we define a function w : ( , ∞ ) → [ , ∞ ) by w ( ξ ) = sup m θ m k ( c ψ m ) ξ k H , then theseestimates imply that w ( ξ ) decays rapidly in ξ in the sense of Section 7. We furtherdefine functions v n , m : ( , ∞ ) → [ , ∞ ) by v n , m ( ξ ) = k L ξ ( c ψ m ) ξ k H , and we noticethat v n , m and w satisfy the hypotheses of Proposition 7.5 and Proposition 7.6. Con-sequently, for any c >
0, the sequence { t n } n ∈ N defined by t n = sup { v n , m ( ξ ) : n − α ≤ ξ < ∞ and m < c log ( n ) } decays rapidly in n .Fix n ∈ N and let m be the largest integer such that m < c log ( n ) , where c = k / ( − log ( θ )) ; in particular n − k = θ c log ( n ) < θ m ≤ θ c log ( n ) − = θ − n − k . UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 27
By Proposition 10.3-(iv), we get | cov ( Φ ◦ F n , ψ ) | ≤ | cov ( Φ m ◦ ( F + ) n , ψ m ) | + K θ − n − k , hence it suffices to bound the first summand in the right-hand side above. ByProposition 4.6, we have | cov ( Φ m ◦ ( F + ) n , ψ m ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z X Z { } ( L n ξ ( c ψ m ) ξ )( x ) d ( η m ) x ( ξ ) d µ ( x ) − ν av ( Φ m ) ν ( ψ m ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z X Z ( , n − α ) ( L n ξ ( c ψ m ) ξ )( x ) d ( η m ) x ( ξ ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) Z X Z [ n − α , ∞ ) ( L n ξ ( c ψ m ) ξ )( x ) d ( η m ) x ( ξ ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) . The last summand in the right-hand side above is bounded by Mt n , hence decaysrapidly. The first term, by Lemma 8.1, is bounded by C k Φ m k G + ( Max ( ψ m ) + Lip ( ψ m )) δ n ≤ CM ( Φ )( M ( ψ , ) + L ( ψ , )) n k δ n , which decays rapidly as well. Finally, for the second term, Lemma 8.2 implies (cid:12)(cid:12)(cid:12)(cid:12) Z X Z ( , n − α ) ( L n ξ ( c ψ m ) ξ )( x ) d ( η m ) x ( ξ ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M ( ψ , ) LF ( Φ m , n − α ) . In order to coclude the proof of Theorem 3.8, it suffices to establish the followinglemma.
Lemma 10.4.
With the notation above, for any r > we have | LF ( Φ m , r ) − LF ( Φ , r ) | ≤ M ( Φ ) θ − n − k . Proof.
Note that the measure associated to ( Φ ◦ F m )( x , · ) is e − i ξ f m ( x ) d η σ m x ( ξ ) ,whose variation is | η σ m x | . Let us denote R = ( − r , r ) \ { } ⊂ R . Then, by Proposi-tion 10.3-(ii), | LF ( Φ m , r ) − LF ( Φ , r ) | = (cid:12)(cid:12)(cid:12)(cid:12) Z X | ( η m ) x | ( R ) d µ ( x ) − Z Σ | η x | ( R ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z Σ | η σ m x | ( R ) d µ ( x ) − Z Σ | η x | ( R ) d µ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) + max x ∈ X k ( η m ) x − e − i ξ f m ( x ) η σ m x k TV = max x ∈ X k Φ m ( x , · ) − Φ ◦ F m ( x , · ) k ≤ M ( Φ ) θ m ≤ M ( Φ ) θ − n − k . (cid:3) A PPENDIX
A. A
CCESSIBILITY AND SYMBOLIC DYNAMICS
Let A : M → M be a diffeomorphism and let Ω ⊂ M be a transitive uniformlyhyperbolic subset. One can construct a Markov partition on Ω and use it to definesymbolic dynamics. That is, there is a subshift of finite type σ : Σ → Σ and acontinuous surjective map π : Σ → Ω such that A ◦ π = π ◦ σ .Let f : M → R be a continuous function which defines a skew product F ( x , t ) = ( A ( x ) , t + f ( x )) on M × R . This function then also defines a “symbolic skew product” F sym ( x , t ) = ( σ ( x ) , t + f sym ( x )) on Σ × R where f sym : Σ → R is given by f sym = f ◦ π . Question A.1.
If F | Ω × R is accessible, does it follow that F sym is accessible? At first glance, the question might seem easy to answer, but there are some subtleissues here. As π is uniformly continuous, if points x and y lie on the same stablemanifold in Σ , then they project down to points π ( x ) and π ( y ) lying on the samestable manifold in M . However, one can construct examples where π ( x ) and π ( y ) lie on the same stable manifold, but σ n ( x ) and σ n ( y ) stay far apart for all n ∈ Z .Hence, not all us -paths in M lift to us -paths in Σ . Despite this, we can establishaccessibility of F sym in certain settings.We will adopt the notation used in Bowen’s book on the subject [6]. In particularrecall that if p and q are in the same rectangle, then [ p , q ] is the intersection of thelocal stable manifold of p with the local unstable manifold of q . If x = { x n } and y = { y n } are elements of the symbolic dynamics with the same “zeroth” symbol x = y , then [ x , y ] = z = { z n } is defined by z n = x n for n ≥ z n = y n for n ≤ π ([ x , y ]) = [ π ( x ) , π ( y )] .For p and q in Ω , if q ∈ W sA ( p ) , define ∆ s ( p , q ) = ∑ ∞ n = f ( A n q ) − f ( A n p ) and notethat ( p , s ) ∈ W sF ( q , t ) if and only if t − s = ∆ s ( p , q ) . If q ∈ W uA ( p ) , define ∆ u ( p , q ) analogously. For p and q in the same rectangle, define h ( p , q ) = ∆ s ( p , [ p , q ]) + ∆ u ([ p , q ] , q ) + ∆ s ( q , [ q , p ]) + ∆ u ([ q , p ] , p ) . That is, h ( p , q ) measures the height of the “Brin quadrilateral” that has p and q astwo of its four vectices.Note that h is continuous and if q is on the local stable or manifold manifold of p , then h ( p , q ) = ∆ s sym , ∆ u sym , and h sym using the same formulas, but with f sym in the placeof f . Then on the cylinder C i ⊂ Σ consisting of the sequences whose zeroth symbolcorresponds to R i , the function h sym : C i × C i → R is continuous and h sym ( x , y ) = h ( π ( x ) , π ( y )) . Proposition A.2.
If p ∈ R i and γ : [ , ] → R i is a continuous curve such thath ( p , γ ( )) is zero and h ( p , γ ( )) is non-zero, then F sym is accessible.Proof. Since h and γ are continuous, I = h ( p , γ ([ , ])) is a positive length intervalcontaining zero. For any t ∈ [ , ] , using the properties of symbolic dynamics wemay find elements x , y ∈ C i such that π ( x ) = p and π ( y ) = γ ( t ) . This implies that h sym ( C i × C i ) contains I . From this, one can show that F sym has an open accessibil-ity class and then use this to conclude that F sym is accessible. (cid:3) Corollary A.3.
If A : M → M is an Anosov diffeomorphism and Ω = M, then for a C -open and C r -dense ( ≤ r < ∞ ) set of choices of f : M → R , the correspondingF sym is accessible.Proof. Choose a rectangle R i in a partition. Since A is Anosov and R i has interior,there is a periodic point p in the interior of R i . Adapting the proof of the “Unweav-ing Lemma”, that is [30, Lemma A.4.3], we can make a small perturbation to anystarting f in a small neighbourhood in order to find a point q with h ( p , q ) =
0. Wecan then define a path γ : [ , ] → R i from p to q and apply Proposition A.2. (cid:3) Corollary A.4.
If A : T → T is an Anosov diffeomorphism, then F is accessibleif and only is F sym is accessible.Proof. Here, A is topologically conjugate to a linear map [16], and so we assume A itself is linear. In this setting, we can find a Markov partition where the interior ofeach rectangle is homeomorphic to a disc and its boundary consists of two stablecurves and two unstable curves. For such a construction, see for instance [27,Section 8]. If F sym is not accessible, then for each rectangle R i the stable and UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 29 unstable directions of F are jointly integrable inside the region R i × R ⊂ M × R .This region is therefore foliated by C surfaces (with boundary) tangent to E s ⊕ E u .As the rectangles meet along stable and unstable curves, we can “glue together” theleaves of neighbouring rectangles to produce a foliation on all of M × R tangent to E s ⊕ E u . (cid:3) Remark A.5.
The proof above is highly specific to the 2-torus. Bowen showed inhigher dimensions that the boundaries of the rectangles are not smooth [5] . More-over, for the standard constuction of a Markov partition of an Anosov diffeomor-phism, it is not clear in general whether the rectangles even have finitely manyconnected components.
We now consider accessibility in the setting of hyperbolic attractors.
Corollary A.6.
If A : M → M is an Axiom A diffeomorphism and Ω ⊂ M is anattractor or repellor, then for a C -open and C r -dense ( ≤ r < ∞ ) set of choicesof f : M → R , the corresponding F sym is accessible.Proof. We will assume Ω is an attractor so that for each p ∈ Ω , the unstable mani-fold W uA ( p ) is a connected immersed submanifold lying entirely within Ω .As shown in [21], there is a neighbourhood of Ω on which one can define in-variant stable and unstable foliations. We may find a periodic point p and a smallneighbourhood U of p , such that every unstable manifold intersects U in a path-connected set.As in the proof of Corollary A.3 above, one can adapt the “Unweaving Lemma”to perturb f and find p and q with h ( p , q ) =
0. We define γ : [ , ] → R i to be a pathfrom [ p , q ] to q and then apply Proposition A.2. (cid:3) A PPENDIX
B. P
ROOFS OF L EMMAS
AND
Proof of Lemma 3.2.
For any fixed R >
0, by the invariance of the GibbsMeasure with respect to the dynamics, we have12 R Z Σ × [ − R , R ] Φ ( x , r ) d ν ( x , r ) = R Z Σ × [ − R , R ] Φ ( σ x , r ) d ν ( x , r ) , therefore12 R (cid:12)(cid:12)(cid:12)(cid:12) Z Σ × [ − R , R ] ( Φ ◦ F − Φ )( x , r ) d ν ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ R (cid:12)(cid:12)(cid:12)(cid:12) Z Σ (cid:18) Z R + f ( x ) − R + f ( x ) Φ ( σ x , r ) d r (cid:19) d µ ( x ) − Z Σ × [ − R , R ] Φ ( σ x , r ) d ν ( x , r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ R k f k ∞ k Φ k ∞ . The last term above converges to zero for R → ∞ , hence the limitlim R → ∞ R Z Σ × [ − R , R ] Φ ◦ F ( x , r ) d ν ( x , r ) exists and equals ν av ( Φ ) .B.2. Proof of Lemma 3.5.
By definition, we can write Φ ( x , r ) = Z ∞ − ∞ e − ir ξ d η x ( ξ ) = η x ( { } ) + Z R \{ } e − ir ξ d η x ( ξ ) . We have to show that the limit (3) exists; in order to do this, we prove that for any x ∈ Σ we have(12) lim R → ∞ R Z R − R Φ ( x , r ) d r = η x ( { } ) , from which the claim follows. For any R > R Z R − R Φ ( x , r ) d r = R Z R − R (cid:18) η x ( { } ) + Z R \{ } e − ir ξ d η x ( ξ ) (cid:19) d r = η x ( { } ) + Z R \{ } (cid:18) R Z R − R e − ir ξ d r (cid:19) d η x ( ξ )= η x ( { } ) + Z R \{ } e iR ξ − e − iR ξ iR ξ d η x ( ξ ) = η x ( { } ) + Z R \{ } sin ( R ξ ) R ξ d η x ( ξ ) . We have that | sin ( R ξ ) / ( R ξ ) | ≤ ∈ L ( | η x | ) , since the total variation of η x is finite) and sin ( R ξ ) / ( R ξ ) converges to 0 for all ξ =
0. Lebesgue theorem yields(12), which completes the proof.B.3.
Proof of Lemma 3.7.
Let us show that for all ε > η (cid:18) R \ (cid:20) − ε , ε (cid:21)(cid:19) ≤ ε Z ε − ε Φ ( ) − Φ ( r ) d r Then, simply by choosing ε = / K , we conclude η ( R \ [ − K , K ]) ≤ K Z / K − / K | Φ ( ) − Φ ( r ) | d r ≤ LK Z / K − / K | r | d r = LK . Fix ε >
0; then12 ε Z ε − ε Φ ( ) − Φ ( r ) d r = ε Z ε − ε Φ ( ) − (cid:18) Z ∞ − ∞ e − i ξ r d η ( ξ ) (cid:19) d t = Φ ( ) − Z ∞ − ∞ Z ε − ε e − i ξ r ε d r ! d η ( ξ ) = Z ∞ − ∞ − (cid:18) Z ε cos ( ξ r ) ε d r (cid:19) d η ( ξ )= Z ∞ − ∞ − sin ( εξ ) εξ d η ( ξ ) = Z [ − ε , ε ] − sin ( εξ ) εξ d η ( ξ ) + Z R \ [ − ε , ε ] − sin ( εξ ) εξ d η ( ξ ) . Since sin ( x ) / x ≤ max { , / | x |} for all x ∈ R , we get12 ε Z ε − ε Φ ( ) − Φ ( r ) d r ≥ Z R \ [ − ε , ε ] − sin ( εξ ) εξ d η ( ξ ) ≥ Z R \ [ − ε , ε ] − | εξ | d η ( ξ ) . Clearly, R \ (cid:2) − ε , ε (cid:3) = n ξ ∈ R : 1 − | εξ | > o , thus, by Chebyshev inequality, η (cid:18) R \ (cid:20) − ε , ε (cid:21)(cid:19) ≤ Z R \ [ − ε , ε ] − | εξ | d η ( ξ ) ≤ ε Z ε − ε Φ ( ) − Φ ( r ) d r , which proves the initial claim.B.4. Proof of Lemma 3.1.
Assume that f is cohomologous to zero, namely thereexists a measurable function w such that f = w ◦ σ − w . By Liv˘sic Theorem, wecan assume that w is continuous. We claim that for any x ∈ Σ , all the points thatcan be reached by an su -path from ( x , w ( x )) ∈ Σ × R are contained in the graph of w , i.e. in the set G ( w ) : = { ( y , w ( y )) : y ∈ Σ } , which will give a contradiction withthe accessibility assumption. UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 31
Let ( x , w ( x )) ∈ G ( w ) ; we now show that the whole stable set W s ( x , w ( x )) is fullycontained in G ( w ) . Let ( y , s ) ∈ W s ( x , w ( x )) ; then, y ∈ W s ( x ) and s − w ( x ) = lim n → ∞ f n ( x ) − f n ( y ) = lim n → ∞ ( w ( σ n x ) − w ( x ) − w ( σ n y )+ w ( y )) = − w ( x )+ w ( y ) , that is s = w ( y ) . This implies W s ( x , w ( x )) ⊂ G ( w ) . An analogous argument showsthat W u ( x , w ( x )) ⊂ G ( w ) .A PPENDIX
C. P
ROOFS OF L EMMA
AND P ROPOSITION
Proof of Lemma 10.2.
In this proof, we will enounter another loss in regular-ity, namely we will need to replace θ with a different one to make the observablesLipschitz: again, this is not an issue and we will freely do so.Let ψ ∈ L and define ψ h ( x , r ) = ψ ( x , r − h ( x )) , where h ∈ F θ . By definition,it is clear that ψ h is well-defined from Σ to S , and, by the Fubini-Tonelli Theorem, ν ( ψ ) = ν ( ψ h ) . The only thing to check is that it is a Lipschitz map.Let a , ℓ be non negative integers, and fix x , y ∈ Σ . We need to estimate k ψ h ( x ) − ψ h ( y ) k a ,ℓ : = sup r ∈ R | r | a | ∂ ℓ ψ h ( x , r ) − ∂ ℓ ψ h ( y , r ) | . By definition, for any r ∈ R , we have | r | a | ∂ ℓ ψ h ( x , r ) − ∂ ℓ ψ h ( y , r ) | ≤| r | a | ∂ ℓ ψ ( x , r − h ( x )) − ∂ ℓ ψ ( x , r − h ( y )) | + | r | a | ∂ ℓ ψ ( x , r − h ( y )) − ∂ ℓ ψ ( y , r − h ( y )) | Let us consider the two summands separately. For the fisrt one, we have | r | a | ∂ ℓ ψ ( x , r − h ( x )) − ∂ ℓ ψ ( x , r − h ( y )) | ≤ | r | a | ∂ ℓ + ψ ( x , r − h ( x )+ u ) |·| h ( x ) − h ( y ) | , for some | u | ≤ h ( x ) − h ( y ) ≤ k h k ∞ . If | r | ≤ k h k ∞ , we have | r | a | ∂ ℓ ψ ( x , r − h ( x )) − ∂ ℓ ψ ( x , r − h ( y )) | ≤ ( k h k ∞ ) a k ψ ( x ) k ,ℓ + | h | θ d θ ( x , y ) , otherwise, if | r | > k h k ∞ , then | r | a | ∂ ℓ ψ ( x , r − h ( x )) − ∂ ℓ ψ ( x , r − h ( y )) | ≤ k ψ ( x ) k a ,ℓ + | h | √ θ d √ θ ( x , y ) . In both cases, the first term satisfies a Lipschitz bound, independent of r . For thesecond term, if | r | ≤ k h k ∞ , then | r | a | ∂ ℓ ψ ( x , r − h ( y )) − ∂ ℓ ψ ( y , r − h ( y )) | ≤ ( k h k ∞ ) a k ψ ( x ) − ψ ( y ) k ,ℓ , and the Lipschitz bound follows from the assumption on ψ ; otherwise, if | r | > k h k ∞ , then | r | a | ∂ ℓ ψ ( x , r − h ( y )) − ∂ ℓ ψ ( y , r − h ( y )) | ≤ k ψ ( x ) − ψ ( y ) k a ,ℓ , and again the conclusion follows from the assumption on ψ . This concludes theproof of the claims on ψ .Let now Φ ∈ G . Then, for any x ∈ Σ , the function Φ h ( x )( r ) = Φ ( x , r − h ( x )) isthe Fourier-Stieltjes transform of the measure d ( η h ) x = e i ξ h ( x ) d η x . In particular,the variation are the same | ( η h ) x | = | η x | and, from Lemma 3.5, it also followsthat ν av ( Φ h ) = ν av ( Φ ) . Again, the only claim left to be shown is the Lipschitzassumption. We will exploit here the tail condition (TC).Fix x , y ∈ Σ . Then, k Φ h ( x ) − Φ h ( y ) k = k ( η h ) x − ( η h ) y k TV ≤ k ( e i ξ h ( x ) − e i ξ h ( y ) ) η x k TV + k η x − η y k TV . The second summand in the right hand-side above satisfies a Lipschitz bound byassumption. We need to verify for the first term. If h ( x ) = h ( y ) , the term is 0 and the proof is complete. Assume h ( x ) = h ( y ) and let I = [ −| h ( x ) − h ( y ) | − / , | h ( x ) − h ( y ) | − / ] . Then we have k ( e i ξ h ( x ) − e i ξ h ( y ) ) η x k TV = Z R | − e i ξ ( h ( x ) − h ( y )) | d | η x | ( ξ )= Z I | − e i ξ ( h ( x ) − h ( y )) | d | η x | ( ξ ) + Z R \ I | − e i ξ ( h ( x ) − h ( y )) | d | η x | ( ξ ) . the first term can be bound by Z I | − e i ξ ( h ( x ) − h ( y )) | d | η x | ( ξ ) ≤ | h ( x ) − h ( y ) | / k η x k TV , and the second by Z R \ I | − e i ξ ( h ( x ) − h ( y )) | d | η x | ( ξ ) ≤ | η x | ( R \ I ) ≤ | h ( x ) − h ( y ) | a / , hence the proof follows from the fact that h is Lipschitz (again, up to possiblyreplacing θ with θ a / if a < Proof of Proposition 10.3.
This section is devoted to the proof of Proposi-tion 10.3. The strategy follows the same lines as in [13] and [26, Proposition 1.2],although there are additional difficulties in showing that the functions Φ and ψ belong to G + and L + respectively.Let α : Σ × R → R be any Lipschitz function, with Lipschitz constant L ( α ) . Fix m ∈ N . The first step is to prove that there exists a function β : Σ × R → R suchthat the function α + = ( α ◦ F m ) + β ◦ F − β depends only on the future coordinates. In other words, we want to show that α ◦ F m is cohomologous to a function defined on X × R . We recall the constructionof β for the reader’s convenience. For any cylinder C n , j : = C − n , o ( x j ) , choose anelement ω n , j ∈ C n , j . Define the element ω n ( x ) ∈ Σ in the following way: ( ω n ( x )) i = ( x i if i ≥ , ω n , ji if i < . Notice that by definition d θ ( ω n ( x ) , x ) ≤ θ n for all x ∈ Σ . Define α ( n ) ( x , r ) : = α ( ω n ( x ) , r + δ n ( x )) , where δ n ( x ) : = f n ( ω n ( x )) − f n ( x ) . Since k δ n k ∞ ≤ | f | θ ( − θ ) − θ n , we have(13) (cid:13)(cid:13)(cid:13) α − α ( n ) (cid:13)(cid:13)(cid:13) ∞ ≤ L ( α ) ( d θ ( ω n ( x ) , x ) + k δ n k ∞ ) ≤ C θ L ( α ) θ n , for a constant C θ = + | f | θ ( − θ ) − . Define β ( x , r ) : = ∞ ∑ n = m ( α ◦ F n − α ( n ) ◦ F n )( x , r ) , which, by (13), is well-defined as a function from Σ × R → R . Then, we can define α + : = α ◦ F m + β ◦ F − β = ∞ ∑ n = m α ( n ) ◦ F n − α ( n ) ◦ F n + , and from its definition, it is clear that α + depends only on the future coordinates;namely, it follows that α ◦ F m is cohomologous to a function defined on X × R .We will show in Lemma C.1 and Lemma C.2 below that if α belongs to L or G , then α + belongs to L + or G + respectively. Assuming these claims, let us first UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 33 finish the proof, namely we show (i) and (iii). There exist Φ + ∈ G + and ψ + ∈ L + which are cohomologous to Φ ◦ F m and ψ ◦ F m respectively. In particular, we have ν ( ψ + ) = ν ( ψ ◦ F m ) = ν ( ψ ) , and ν av ( Φ + ) = ν av ( Φ ◦ F m ) = ν av ( Φ ) . Moreover, from the definitions of ψ + and Φ + and from (13), it follows that k Φ + − Φ ◦ F m k = O ( θ m ) , and similarly for ψ . Therefore, by invariance of the measure ν u under F m , weobtain | cov Φ , ψ ( n ) | = | cov Φ ◦ F m , ψ ◦ F m ( n ) | = | cov Φ , ψ ( n ) | + O ( θ m ) , which concludes the proof. Lemma C.1. If α ∈ L , then α + ∈ L + . For any ℓ ∈ N , there exist constantsM ( A , ℓ ) and L ( A , ℓ ) depending on α and ℓ only such that Max ℓ ( α + ) ≤ M ( α , ℓ ) and Lip ℓ ( α + ) ≤ θ − m L ( α , ℓ ) .Proof. In order to prove the result, it is sufficient to show that for every x ∈ Σ ,the function β ( x , · ) is Schwartz and the functions x ∂ ℓ β ( x , · ) are lipschitz withrespect to the L -norm.Since, for every x ∈ Σ and ℓ ∈ N , the series ∞ ∑ n = m ∂ ℓ α ( σ n x , r + f n ( x )) − ∂ ℓ α ( ω n ( σ n x ) , r + f n ( x ) + δ n ( σ n x )) converges uniformly, the derivative ∂ ℓ β ( x , · ) exists and equal the series above. Wenow show that β ( x , · ) is a Schwartz function for every x ∈ Σ .Let a , ℓ ∈ N . We need to show that r a ∂ ℓ β ( x , r ) is uniformly bounded in r . Tosave notation, let us write x n = σ n x , y n = ω n ( σ n x ) and δ n = δ n ( σ n x ) . We have | r a ∂ ℓ β ( x , r ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) r a ∞ ∑ n = m ∂ ℓ α ( x n , r + f n ( x )) − ∂ ℓ α ( y n , r + f n ( x )) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) r a ∞ ∑ n = m ∂ ℓ α ( y n , r + f n ( x )) − ∂ ℓ α ( y n , r + f n ( x ) + δ n ) (cid:12)(cid:12)(cid:12)(cid:12) Each term in the sum above satisfies a Lipaschitz bound exactly as in the proof ofLemma 10.2. Since the terms d ( x n , y n ) and δ n can be bounded by O ( θ n ) , the seriesabove converge. Therefore, | r a ∂ ℓ β ( x , r ) | is uniformly bounded for all a , ℓ ∈ N ,hence β ( x , · ) is a Schwartz function.The lipschitz bounds on the functions x ∂ ℓ β ( x , · ) with respect to the L -normcan be proved in a similar way and is left as an exercise to the reader: we remarkthat if α is a lipschitz function with constant L ( α ) , then α ◦ F m is lipschitz withconstant L ( α ◦ F m ) ≤ θ − m L ( α ) . (cid:3) We now show the analogous result if we assume that α is a global observable. Lemma C.2. If α ∈ G , then α + ∈ G + , and k α + k G + ≤ k α k G .Proof. It suffices to show that for any x ∈ Σ , β ( x , · ) is the Fourier-Stieltjes trans-form of a complex measure η x and moreover the total variation k η x k TV is uniformlybounded.By definition, β ( x , r ) = lim N → ∞ N ∑ n = m α ( σ n x , r + f n ( x )) − α ( ω n ( σ n x ) , r + f n ( x ) + δ n ( σ n x )) ! . Fix x ∈ Σ and again let us denote x n = σ n x , y n = ω n ( σ n x ) and δ n = δ n ( σ n x ) . For N ≥ m , let ζ N = ζ ( ) N + ζ ( ) N be the complex measure defined byd ζ N ( ξ ) = d ζ ( ) N ( ξ ) + d ζ ( ) N ( ξ ) = N ∑ n = m e i ξ f n ( x ) d η x n ( ξ ) − e i ξ ( f n ( x )+ δ n ) d η y n ( ξ ) , where d ζ ( ) N ( ξ ) = N ∑ n = m e i ξ f n ( x ) d η x n ( ξ ) − e i ξ f n ( x ) d η y n ( ξ ) and d ζ ( ) N ( ξ ) = N ∑ n = m ( e i ξ f n ( x ) − e i ξ ( f n ( x )+ δ n ) ) d η y n ( ξ ) . From the expression for β above, by definition, the Fourier-Stieltjes transforms c ζ N of ζ N converge uniformly to β ( x , · ) . In order to conclude that β ( x , · ) is the Fourier-Stieltjes transform of a complex measure, it is enough to show that the family ofmeasures ζ N is contained in a weakly compact set. We will proceed as in the proofof Lemma 10.2 and use the tightness condition (TC).We will show this separately for ζ ( ) N and ζ ( ) N . For ζ ( ) N , by the lipschitz assump-tion on α , we have k ζ ( ) N k TV = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = m e i ξ f n ( x ) η x n − e i ξ f n ( x ) η y n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) TV ≤ N ∑ n = m k η x n − η y n k TV ≤ k α k G N ∑ n = θ n . The total variation norm is stronger that the weak-convergence topology, hencethe sequence of measures ζ ( ) N converges weakly (since the tails are exponentiallysmall). For the second term, we apply Prokhorov theorem: it suffices to show thatthe sequence ζ ( ) N is uniformly bounded in total variation norm and is tight . Noticethat the variation of ζ ( ) N can be bounded by | ζ ( ) N | ≤ N ∑ n = m | − e i ξδ n || η y n | . Using the tightness condition (TC), k ζ ( ) N k TV = | ζ ( ) N | ( R ) ≤ N ∑ n = m | − e i ξδ n || η y n | (cid:0) [ − θ − n / , θ − n / ] (cid:1) + N ∑ n = m | − e i ξδ n || η y n | (cid:0) R \ [ − θ − n / , θ − n / ] (cid:1) ≤ N ∑ n = m | η y n | ( R ) max ξ ∈ [ − θ − n / , θ − n / ] | − e i ξδ n | ! + N ∑ n = m | η y n | (cid:0) R \ [ − θ − n / , θ − n / ] (cid:1) ≤ k A k G N ∑ n = m k δ n k θ − n / + A k A k G N ∑ n = m θ an / ≤ ( + A ) k A k G N ∑ n = m ( θ n / + θ an / ) . UANTITATIVE GLOBAL-LOCAL MIXING FOR ACCESSIBLE SKEW PRODUCTS 35
This shows that ζ ( ) N is uniformly bounded in total variation. Similarly we provetightness: let us fix ε >
0, and let K = [ − ε − / a , ε − / a ] . Then, | ζ ( ) N | ( R \ K ) ≤ N ∑ n = m | − e i ξδ n || η y n | (cid:0) [ − ε − / θ − n / , ε − / θ − n / ] ∩ ( R \ K ) (cid:1) + N ∑ n = m | − e i ξδ n || η y n | (cid:0) R \ [ − ε − / θ − n / , ε − / θ − n / ] (cid:1) ≤ N ∑ n = m | η y n | ( R \ K ) k δ n k θ − n / ε − / + A ε a / N ∑ n = m θ an / ≤ A ε / N ∑ n = m θ n / + A ε a / N ∑ n = m θ an / . This proves tightness and hence concludes the proof. (cid:3)
Acknowledgements.
P.G. acknowledges the support of the Centro di Ricerca Matem-atica Ennio de Giorgi and of UniCredit Bank R&D group through the â ˘AIJDynam-ics and Information Theory Instituteâ ˘A˙I at the Scuola Normale Superiore. Thisresearch was partially funded by the Australian Research Council.We would like to thank Dima Dolgopyat, Marco Lenci, Federico RodriguezHertz, Omri Sarig for several useful discussions.R
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