Good inducing schemes for uniformly hyperbolic flows, and applications to exponential decay of correlations

Given an Axiom A attractor for a $C^{1+\alpha}$ flow ($\alpha>0$), we construct a countable Markov extension with exponential return times in such a way that the inducing set is a smoothly embedded unstable disk. This avoids technical issues concerning irregularity of boundaries of Markov partition elements and enables an elementary approach to certain questions involving exponential decay of correlations for SRB measures.


Introduction
Statistical properties [12,29,31] of Anosov and Axiom A diffeomorphisms [3,33] were developed extensively in the 1970s.Key tools were the construction of finite Markov partitions [10,32] and the spectral properties of transfer operators [28].In particular, ergodic invariant probability measures were constructed corresponding to any Hölder potential; moreover, it was shown that hyperbolic basic sets for Axiom A diffeomorphisms are always exponentially mixing up to a finite cycle for such measures, see for example [12,22,29].
Still in the 1970s, finite Markov partitions were constructed [11,26] for Anosov and Axiom A flows.This allows us to model each hyperbolic basic set as a suspension flow over a subshift of finite type, enabling the study of thermodynamic formalism (see e.g.[14]) and statistical properties (see e.g.[17,27]).
However, rates of mixing for Axiom A flows are still poorly understood.By [24,30], mixing Axiom A flows can mix arbitrarily slowly.Although there has been important progress starting with [16,18,20], it remains an open question whether mixing Anosov flows have exponential decay of correlations [14].Very recently, this question was answered positively [35] in the case of C ∞ three-dimensional flows.
It turns out that using finite Markov partitions for flows raises technical issues due to the irregularity of their boundaries [5,15,34].Even in the discrete-time setting, it is known that the boundaries of elements of a finite Markov partition need not be smooth [13].In this paper, we propose using the approach of [36] to circumvent such issues at least in the case of SRB measures.In particular, we show that Any attractor for an Axiom A flow can be modelled by a suspension flow over a full branch countable Markov extension where the inducing set is a smoothly embedded unstable disk.The roof function, though unbounded, has exponential tails.
A precise statement is given in Theorem 2.1 below.
Remark 1.1 The approach of Young towers [36] has proved to be highly effective for studying discrete-time examples like planar dispersing billiards and Hénon-like attractors where suitable Markov partitions are not available.However, as shown in the current paper, there can be advantages (at least in continuous time) to working with countable Markov extensions even when there is a well-developed theory of finite Markov partitions.The extra flexibility of Markov extensions can be used not only to construct the extension but to ensure good regularity properties of the partition elements.
As a consequence of Theorem 2.1, we obtain an elementary proof of the following result: Theorem 1.2 Suppose that Λ is an Axiom A attractor with SRB measure µ for a C 1+ flow φ t with C 1+ stable holonomies 1 and such that the stable and unstable bundles are not jointly integrable.Then for all Hölder observables v, w : Λ → R, there exist constants c, C > 0 such that for all t > 0.
Remark 1.3 Joint nonintegrability holds for an open and dense set of Axiom A flows and their attractors, see [19] and references therein.It implies mixing and is equivalent to mixing for codimension one Anosov flows.It is conjectured to be equivalent to mixing for Anosov flows [23].
Remark 1.4 (a) In the case when the unstable direction is one-dimensional and the stable holonomies are C 2 , this result is due to [9,8,4,5].In particular, using the fact that stable bunching is a robust sufficient condition for smoothness of stable holonomies together with the robustness of joint nonintegrability, [4] constructed the first robust examples of Axiom A flows with exponential decay of correlations.The smoothness condition on stable holonomies was relaxed from C 2 to C 1+ in [6] extending the class of examples in [4].This class of examples is extended further by Theorem 1.2 with the removal of the one-dimensionality restriction on unstable manifolds.
(b) There is no restriction on the dimension of unstable manifolds in [8], and it is not surprising that the smoothness assumption on stable holonomies can also be relaxed as in [6].However, there is a crucial hypothesis in [8] on the regularity of the inducing set in the unstable direction which is nontrivial in higher dimensions.
Theorem 1.2 is stated in the special case of Anosov flows in [15].In [15,Appendix] it is argued that at least in the Anosov case the Markov partitions of [26] are sufficiently regular that the methods in [8] can be pushed through.In [5], a sketch is given of how to prove Theorem 1.2 also in the Axiom A case, but the details are not fully worked out.
As mentioned, our approach in this paper completely bypasses such issues since our inducing set is a smoothly embedded unstable disk.Moreover, our method works equally well for Anosov flows and Axiom A attractors.As a consequence, we recover the examples in [15], in particular that codimension one volume-preserving mixing C 1+ Anosov flows are exponentially mixing in dimension four and higher.
The remainder of the paper is organised as follows.In Section 2, we state precisely and prove our result on good inducing for attractors of Axiom A flows.In Section 3, we prove a result on exponential mixing for a class of skew product Axiom A flows, extending/combining the results in [6,8].In Section 4, we complete the proof of Theorem 1.2.

Good inducing for attractors of Axiom A flows
Let φ t : M → M be a C 1+ flow defined on a compact Riemannian manifold (M, d M ), and let Λ ⊂ M be a closed φ t -invariant subset.We assume that Λ is an attracting transitive uniformly hyperbolic set with adapted norm and that Λ is not a single trajectory.In particular, there is a continuous Dφ t -invariant splitting T Λ M = E s ⊕ E c ⊕ E u where E c is the one-dimensional central direction tangent to the flow, and there exists λ ∈ (0, 1) such that |Dφ t v| ≤ λ t |v| for all v ∈ E s , t ≥ 1; |Dφ −t v| ≤ λ t |v| for all v ∈ E u , t ≥ 1.Since the time-s map φ s : Λ → Λ is ergodic for all but countably many choices of s ∈ R [25], we can scale time slightly if necessary so that φ −1 : Λ → Λ is transitive.Then there exists p ∈ Λ such that i≥1 φ −i p is dense in Λ.
We can define (local) stable disks W s δ (y) = {z ∈ W s (y) : d M (y, z) < δ} for δ > 0 sufficiently small for all y ∈ Λ. Define local centre-stable disks W cs δ (y) = |t|<δ φ t W s δ (y).Let Leb and d denote induced Lebesgue measure and induced distance on local unstable manifolds.It is convenient to define local unstable disks W u δ (y) = {z ∈ W u (y) : d(y, z) < δ} using the induced distance.
For δ 0 small, define D = W u δ 0 (p) and D = x∈D W cs δ 0 (x).Define π : D → D such that π|W cs δ 0 (x) ≡ x.Whenever φ n y ∈ D, we set g n y = π(φ n y).We are now in a position to give a precise description of our inducing scheme.In the remainder of this section, we prove Theorem 2.1.Our proof is essentially the same as in [36,Section 6] for Axiom A diffeomorphisms, but we closely follow the treatment in [2] which provides many of the details of arguments sketched in [36].
Choice of constants Choose δ 0 > 0 so that the following bounded distortion property holds: there exists C 1 ≥ 1 so that for every n ≥ 1 and all x, y ∈ Λ with φ n x, φ n y in the same unstable disk such that d(φ j x, φ j y) < 4δ 0 for all 0 ≤ j ≤ n.
By standard results about stable holonomies, π is absolutely continuous and C α for some α ∈ (0, 1) when restricted to unstable disks in D. For δ 0 sufficiently small, there exists C 2 , C 3 ≥ 1 such that for all Lebesgue-measurable subset E ⊂ W u δ 0 (y) ∩ D and all y ∈ Λ, and for all x, y ∈ D with x, y in the same unstable disk such that d(x, y) < 4δ 0 .
Let d u = dim E u and fix L ≥ 3 so that By the local product structure, there exists δ 1 ∈ (0, δ 0 ) such that W cs δ 0 (x) ∩ W u δ 0 (y) consists of precisely one point for all x, y ∈ Λ with d M (x, y) < 4δ 1 .Similarly, there exists δ ∈ (0, δ 1 ) such that W cs δ 1 (x)∩W u δ 1 (y) consists of precisely one point for all x, y ∈ Λ with d M (x, y) < (L+1)δ.Moreover, since local unstable/stable disks have bounded curvature, the intersection point Construction of the partition We consider various small neighbourhoods Fix ε > 0 small (as stipulated in Propositions 2.4 and 2.5 and Lemma 2.9 below).We define sets Y n and functions t n : Y n → N, and R : Y → Z + inductively, with Y n = {R > n}.Define Y 0 = Y and t 0 ≡ 0. Inductively, suppose that Y n−1 = Y \ {R < n} and that t n−1 : Consider the neighbourhood is homeomorphic under g m to k≥n−m+1 I k with outer ring homeomorphic under g m to I n−m+1 , and the union of outer rings is the set {t n = 1}.This picture presupposes Proposition 2.4 below which guarantees that each new generation of collars C n (n) does not intersect the set 1≤m≤n−1 C n−1 (m) of collars in the previous generation.
Proof We argue by contradiction.There is nothing to prove for n = 1.Let n ≥ 2 be least such that the result fails and choose j such that which is a contradiction.This rules out case (i).
In case (ii), choose x ∈ U L−1 nj ∩ ∂A n−1 .We show below that there exists y ∈

∂A (ε)
n−1 such that d(φ n x, φ n y) ≤ ε.In particular, g n x and g n y are well-defined and It remains to verify that there exists y ∈ ∂A is a disjoint union of collars as described in the visualisation above.Hence there exists a collar n−1 and there exists y ∈ S with the desired properties.(The point of the claim is that S lies entirely in Y n−1 .)Note that g n−k maps Q homeomorphically onto the set J = i≥k I i which is an annulus of radial thickness δλ αk .By (2.3), φ n−k maps Q homeomorphically onto a set J = π −1 J of radial thickness at least (C −1 Proof (a) Suppose that t n−1 (y) > 1.Then there exists a collar in Recall that Q is homeomorphic under g n−k to I k .Moreover, g n−k q j lie in distinct components of the boundary of I k , so to be the outer ring of the corresponding collar.Choosing q 1 and q 2 as in part (a) we again obtain a contradiction.Lemma 2.6 There exists a 1 > 0 such that for all n ≥ 1, where (c) Proceeding as in part (b) with U 2 nj \ U 1 nj replaced by U 1 nj , leads to the estimate Corollary 2.7 For all n ≥ 1, Hence by Lemma 2.6(b,c), , proving (a).Similarly, by Lemma 2.6(a), We prove the result by induction.The case n = 0 is trivial since B 0 = ∅.For the induction step from n − 1 to n, we consider separately the cases ). Hence by the induction hypothesis, establishing the result at time n.
Suppose first that V ε ⊂ A n+i for all 0 ≤ i ≤ N. By claim (*), there exists by assumption.This means that V ε ⊃ U L n+i,j for some j.Hence and we are done.In this way, we reduce to the second case where there exists 0 Since V ε \ A n+i = ∅, this means that there exists j so that V ε intersects U 2 n+i,j .Hence we can choose a Hence, using the local product structure and definition of δ 1 , we can define b verifying claim (**).We are now in a position to complete the proof of the lemma.Let n ≥ 1, and let Z ⊂ φ n A n−1 be a maximal set of points such that the balls Let z ∈ Z and let U z = U 1 n+i,j be as in claim (**).In particular, . .
Finally, the sets U z are connected components of 0≤i≤N {R = n + i} lying in distinct disjoint sets φ −n W u δ 0 (z).Hence as required.
We can now complete the proof of Theorem 2.1.
Proof By Corollary 2.8 and Lemma 2.9, 2 ) 1/N and the result follows.

Exponential decay of correlations for flows
In this section, we consider exponential decay of correlations for a class of uniformly hyperbolic skew product flows satisfying a uniform nonintegrability condition, generalising from C2 flows as treated in [8] to C 1+α flows.In doing so, we remove the restriction in [9,6] that unstable manifolds are one-dimensional.
The arguments are a straightforward combination of those in [6,8].We follow closely the presentation in [6], with the focus on incorporating the ideas from [8] where required.
Quotienting by stable leaves leads to a class of semiflows considered in Subsection 3.1.The flows are considered in Subsection 3.2.
The current section is completely independent from Section 2, so overlaps in notation will not cause any confusion.Suppose that F : U ∈P U → Y is C 1+α on each U ∈ P and maps U diffeomorphically onto Y .Let H = {h : U → Y : U ∈ P} denote the family of inverse branches, and let H n denote the inverse branches for F n .We say that F is a C 1+α uniformly expanding map if there exist constants

C 1+α uniformly expanding semiflows
where |ψ| α = sup y =y ′ |ψ(y) − ψ(y ′ )|/d(y, y ′ ) α .Under these assumptions, it is standard [1] that there exists a unique F -invariant absolutely continuous measure µ.The density dµ/d Leb is C α , bounded above and below, and µ is ergodic and mixing.We consider roof functions r : U ∈P U → R + that are C 1 on partition elements U with inf r > 0. Define the suspension Y r = {(y, u) ∈ Y × R : 0 ≤ u ≤ r(y)}/ ∼ where (y, r(y)) ∼ (F y, 0).The suspension semiflow F t : Y r → Y r is given by F t (y, u) = (y, u+t) computed modulo identifications, with ergodic invariant probability measure µ r = (µ × Lebesgue)/r where r = Y r dµ.We say that F t is a C 1+α uniformly expanding semiflow if F is a C 1+α uniformly expanding map and we can choose C 1 from condition (i) and ε > 0 such that for h 1 , h 2 ∈ H n .We require the following uniform nonintegrability condition [8, Equation (6.6)]: (UNI) There exists E > 0 and h 1 , h 2 ∈ H n 0 , for some sufficiently large n 0 ≥ 1, with the following property: There exists a continuous unit vector field ℓ : (The requirement "sufficiently large" can be made explicit as in [6, Equations (2.1) to (2.3)].)From now on, n 0 , h 1 and h 2 are fixed.
where ∂ t denotes differentiation along the semiflow direction.
We can now state the main result in this section.
Theorem 3.1 Suppose that F t : Y r → Y r is a C 1+α uniformly expanding semiflow satisfying (UNI).Then there exist constants c, C > 0 such that In the remainder of this subsection, we prove Theorem 3.1.For s ∈ C, let P s denote the (non-normalised) transfer operator We introduce the family of equivalent norms The proof now proceeds exactly as for [6, Proposition 2.5] (with R, h ′ and |x − y| changed to r, det Dh and d(x, y)).
Only step (3) requires any change from the argument in [6, Lemma 2.9].We provide here the modified argument.Approximate the continuous unit vector field ℓ : R m → R m in (UNI) by a smooth vector field ℓ : R m → R m with |ℓ(x)| ≤ 1 for all x ∈ R m .By condition (iii 1 ), the approximation can be chosen close enough that Let g : [0, ∆/|b|] → R m be the solution to the initial value problem Proof This is unchanged from [6, Lemma 2.12].Lemma 3.9 (L 2 contraction) There exist ε, β ∈ (0, 1) such that where χ m = χ(b, u m , v m ).It is immediate from the definitions that (u 0 , v 0 ) ∈ C b , and it follows from Lemma 3.8 that (u m , v m ) ∈ C b for all m.Hence inductively the χ m are well-defined as in Corollary 3.6.We proceed as in [6, Lemma 2.13] in the following steps.
We say that f : X → X is a C 1+α uniformly hyperbolic skew product if F : Y → Y is a C 1+α uniformly expanding map satisfying conditions (i) and (ii) as in Section 3.1, with absolutely continuous invariant probability measure µ, and moreover (v) There exist constants C > 0, γ 0 ∈ (0, 1) such that d(f n (y, z), f n (y, z ′ )) ≤ Cγ n 0 d(z, z ′ ) for all y ∈ Y , z, z ′ ∈ Z.Let π s : X → Y be the projection π s (y, z) = y.This defines a semiconjugacy between f and F and there is a unique f -invariant ergodic probability measure µ X on X such that π s * µ X = µ.Suppose that r : U ∈P U → R + is C 1 on partition elements U with inf r > 0. Define r : X → R + by setting r(y, z) = r(y).Define the suspension X r = {(x, u) ∈ X × R : 0 ≤ u ≤ r(x)}/ ∼ where (x, r(x)) ∼ (f x, 0).The suspension flow f t : X r → X r is given by f t (x, u) = (x, u + t) computed modulo identifications, with ergodic invariant probability measure µ r X = (µ X × Lebesgue)/r.We say that f t is a C 1+α uniformly hyperbolic skew product flow provided f : X → X is a C 1+α uniformly hyperbolic skew product as above, and r : Y → R + satisfies conditions (iii) and (iv) as in Section 3.1.If F : Y → Y and r : Y → R + satisfy condition (UNI) from Section 3.1, then we say that the skew product flow f t satisfies (UNI). Define Define F α,k (X r ) to consist of functions with v α,k = k j=0 ∂ j t v α < ∞ where ∂ t denotes differentiation along the flow direction.
We can now state the main result in this section.Given v ∈ L 1 (X r ), w ∈ L ∞ (X r ), define the correlation function Theorem 3.11 Assume that f t : X → X is a C 1+α hyperbolic skew product flow satisfying the (UNI) condition.Then there exist constants c, C > 0 such that |ρ v,w (t)| ≤ Ce −ct v α,1 w α,1 , for all t > 0 and all v, w ∈ F α,1 (X r ) (alternatively all v ∈ F α,2 (X r ), w ∈ F α (X r )).
Next, we find a more convenient expression for D 0 in terms of r and f .Note that for any x ∈ X, there exists N(x) ∈ Z + (the number of returns to X up to time r(x)) such that r(x) = N (x)−1 ℓ=0 τ (g ℓ x), f (x) = g N (x) x.
Corresponding to the partition P of Y , we define the collection P = { Ū × Z : U ∈ P} of closed subsets of X. Suppose that x, x ′ ∈ V 0 , V 0 ∈ P, with x ′ ∈ W u (x).The induced map f : X → X need not be invertible since it is not the first return to X.However, we may construct suitable inverse branches z j , z ′ j of x, x ′ as follows.Set z 0 = x, z ′ 0 = x ′ .Since f is transitive and continuous on closures of partition elements, there exists V 1 ∈ P and z 1 ∈ V 1 such that f z 1 = z 0 .Since F is full-branch, f (W u (z 1 ) ∩ V 1 ) ⊃ W u (z 0 ), so there exists z ′ 1 ∈ W u (z 1 ) ∩ V 1 such that f z ′ 1 = z ′ 0 .Inductively, we obtain V n ∈ P and z j , z ′ j ∈ V n with z ′ j ∈ W u (z j ) such that f z j = z j−1 and f z ′ j = z ′ j−1 .By construction, z j−1 = f z j = g N (z j ) z j .Hence z j = g −(N (z 1 )+•••+N (z j )) x and r(z j ) = N (z j )−1 ℓ=0 τ (g ℓ g −(N (z 1 )+•••+N (z j )) x) = N (z 1 )+•••+N (z j ) ℓ=N (z 1 )+•••+N (z j−1 )+1 τ (g −ℓ x).
A similar expression holds for r(z ′ j ).Hence r(z j ) − r(z ′ j ) .
We are now in a position to complete the proof of the lemma, showing that if (UNI) fails, then D ≡ 0. To do this, we make use of [8,Proposition 7.4] (specifically the equivalence of their conditions 1 and 3).Namely, the failure of the (UNI) condition in Section 3.For x, x ′ ∈ V 0 , V 0 ∈ P, with x ′ ∈ W u (x), it follows that n j=1 r(z j ) − r(z ′ j )} = ξ(x) − ξ(x ′ ) − ξ(z n ) + ξ(z ′ n ).
Proof of Theorem 1.2 By Proposition 4.3 and Lemma 4.4, f t is a C 1+α uniformly hyperbolic flow satisfying (UNI).The result for C 1+α observables follows from Theorem 3.11.As in [18], the result follows from a standard interpolation argument (see also [6,Corollary 2.3]).

1 .
Hence part (c) follows from Lemma 2.6(b) and part (b).Similarly, A n = A n ∩ A n−1 ∪ B n−1 and part (d) follows from Lemma 2.6(a) and part (a).

Fix
α ∈ (0, 1).Let Y ⊂ R m be an open ball 2 in Euclidean space with Euclidean distance d.We suppose that diam Y = 1.Let Leb denote Lebesgue measure on Y .Let P be a countable partition mod 0 of Y consisting of open sets.

where c 1 =Lemma 3 . 8 (
c 2 e −K ′ .Since the sets B i ⊂ Y are disjoint, Invariance of cone) There is a constant C 4 depending only on C 1 , C 2 , |f −1 0 | ∞ and |f 0 | α such that the following holds: For all (u, v) ∈ C b , we have that

3. 2 C
1+α uniformly hyperbolic skew product flows Let X = Y × Z where Y is an open ball of diameter 1 with Euclidean metric d Y and (Z, d Z ) is a compact Riemannian manifold.Define the metric d((y, z), (y 1 means that we can write r = ξ • F − ξ + ζ on Y where ξ : Y → R is continuous (even C 1 ) and ζ is constant on partition elements U ∈ P. Extending ξ and ζ trivially to X = Y × Z, we obtain that r = ξ • f − ξ + ζ on X where ξ : X → R is continuous and constant on stable leaves, and ζ is constant on elements V ∈ P. In particular, n j=1 r(z j ) = n j=1 ξ(z j−1 − ξ(z j ) + ζ(z j ) = ξ(x) − ξ(z n ) + n j=1 ζ(z j ).
1); (ii) Each connected component of {R = n} is mapped by φ n into D and mapped homeomorphically by g n onto Y .Let P be the partition of Y consisting of connected components of {R = n} for n ≥ 1. (It follows from Theorem 2.1(i) that P is a partition of Y mod 0.) Define F : Y → Y , F = g R = π • φ R .Note that F is locally the composition of a time-R map φ R (where R is constant on each partition element) with a centre-stable holonomy.Since centre-stable holonomies are Hölder continuous, it follows that F maps partition elements U ∈ P homeomorphically onto Y and that F | U : U → Y is a bi-Hölder bijection.If moreover, the centre-stable holonomies are C 1 , then the partition elements are diffeomorphic to disks.