Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Given any smooth Anosov map we construct a Banach space on which the associate transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.


Introduction
Starting with the paper [BKL], there has been a growing interest in the possibility to develop a functional analytic setting allowing the direct study of the transfer operator of a hyperbolic dynamical system. The papers [GL,GL1,BT,BT1,B1,B2,B3,B4,T1] have now produced quite satisfactory results for the case of Anosov diffeomorphisms (or, more generally, for uniformly hyperbolic basic sets). Important results, although the theory is not complete yet, have been obtained for flows [L,BuL,BuL2,GLP,FT2,DyZ,D17], group extensions and skew products ( [F11,AGT]). Moreover, recently it has been unveiled a strong relation with techniques used in semiclassical analysis (e.g. see [FR,FRS,FT1,FT2,DyZ]). Also, one should mention the recent discovery of a deep relation with the theory of renormalization of parabolic systems [GL19]. In addition, such an approach has proven very effective in the study of perturbation of dynamical systems [KL1,KL3] and in the investigation of limit theorems [G10]. At the same time [KL2,KL4] has shown that this strategy can be extended to the infinite dimensional setting (limited to the case of piecewise expanding maps). However, there has been no progress in applying it to coupled lattices of Anosov maps. Moreover, only partial progresses have been accomplished in extending such an approach to partially hyperbolic maps [T2] and to piecewise smooth uniformly hyperbolic systems [DL,BG1,BG2,DZ1,DZ2,DZ3,BaL,BDL]. The recent book [B5] provides an extensive account and a thorough illustration of the topic.
The present paper is motivated by the current shortcomings in the applications of the functional analytic strategy to piecewise smooth hyperbolic maps. Indeed, while in two dimensions the approach can be applied to a large class of systems, in higher dimensions it is limited to the case in which the map is well behaved up to and including the boundary [BG1,BG2]. In the case of piecewise expanding maps the latter problems are dealt with by the using different Banach spaces. In particular, a huge class of piecewise expanding maps can be treated by using the space of functions of bounded variation (BV ) or their straightforward generalisations [Sa00,Li13a,Bu13,Li13b]. It is thus natural to construct Banach spaces that generalize BV and are adapted to the study of the transfer operator associated with hyperbolic maps. Unfortunately, none of the Banach spaces proposed in the literature for the study of the transfer operator associated with Anosov diffeomorphisms, or piecewise Anosov, reduces exactly to BV when the stable direction is absent.
The purpose of this paper is to correct this state of affairs introducing a template for Banach spaces with the above property. We will apply it to the case of smooth Anosov diffeomorphisms. Although for such examples this provides limited new information, it shows that the proposed space is well adapted to the hyperbolic structure, hence there is a concrete hope that this space can be adapted to study general Anosov piecewise maps and Anosov coupled map lattices in a unified setting. A substantial amount of work is still needed to find out if such a hope has some substance or not. Nevertheless, the present arguments are worth presenting since they are remarkably simple and natural.
An additional fact of interest in the present paper is the characterization of invariant foliations and, more generally, the method used to study the evolution of foliations under the dynamics. It is well known that the stable foliation is only Hölder, although the leaves of the foliations enjoy the same regularity as the map. Nevertheless, a fundamental discovery by Anosov is that the holonomy associated to the foliation is absolutely continuous and the Jacobian is Hölder. The establishing of this fact is not trivial and, especially in the discontinuous case, entails a huge amount of work [KS]. Here we show that the properties of such foliations can be characterized infinitesimally, hence considerably simplifying their description, see Definition 3. In particular, given a foliation F , the Jacobian J F of the associated Holonomy can be seen as a quantity produced by a flow, of which we control the generator H F . See Lemma B.7 for a precise explanation of this fact. We believe this point of view will be instrumental in treating discontinuous maps.
The structure of the paper is as follows: Section 2 contains the definition of the Banach space and the statement of the main theorem (Theorem 2.1). Section 3 contains the usual Lasota-Yorke estimate. While section 4 contains the estimate on the essential spectrum of the operator. Section 5 contains some comments on the peripheral spectrum. Appendix A reminds to the reader some convenient properties of C r norms. Appendix B establishes various properties of the foliations of Anosov maps that should be folklore among experts, but we could not locate anywhere (in particular the smoothness of the Jacobian of the stable holonomy along stable leaves). Moreover, as previously mentioned, such properties are express totally in local terms, contrary to the usual approach. Finally, Appendix C contains few technical estimates on the test functions.
Notation. In this paper we will use C ♯ to designate a constant that depends only on the map T and on the choice of coordinates, but whose actual value is irrelevant to the tasks at hand. Hence, the value of C ♯ can change from one occurrence to the next. While we will use C n , c n for constants that may depend on the value of n and, more generally, C a,b,... , c a,b,... for constants that depend on the parameters a, b, . . . .

The Banach space
Our goal is to develop a space in the spirit of BV for the study of the statistical properties of a dynamical system (M, T, µ) where M is compact a C r manifold, T is uniformly hyperbolic and µ is the SRB measure. 1 Let us be more precise.
2.1. The phase space. Let r ≥ 2 be an integer and M be a C r d-dimensional compact manifold where the differentiable structure is the one induced by the atlas To be more precise we consider a fixed smooth partition of unity {ϑ i } subordinated to the cover {V i }. We then define a smooth volume form ω by . From now on all the integral will be with respect to such a form although we will not specify it explicitly.
2.2. The map and the cones. We consider an Anosov diffeomorphism T ∈ Diff r (M ). That is, there exists λ > 1, ν ∈ (0, 1), c 0 ∈ (0, 1) and a continuous cone field (stable cone) In higher dimensions a cone may have many geometric shapes. It is convenient, and useful, to ask that they be subsets K of the Grassmannian. More precisely, we can assume, without loss of generality, that, for each ξ ∈ V i and calling M(d u , d s ) the set of d u × d s matrices, where U is any d u × d s matrix. Then the strict cone field invariance reduces to the existence of η ∈ (0, 1) such that 1 Of course, there are many other functional spaces to analyse such maps (e.g. see [B5]), however we restrict to this class of maps to illustrate the construction of the space in the simplest possible form. 2 Here the norm is defined by some smooth Riemannian structure, the actual choice of such a structure will be irrelevant in the following, it will just affect the constants.
2.3. Transfer Operator. We are interested in studying the statistical properties of the above systems. One key tool used to such an end is the Transfer Operator: Accordingly, for each n ∈ N, It is then clear that the behaviour of the integrals on the left of the above equation can be studied if one understands the spectrum of L. Obviously such a spectrum depends on the space on which the operator is defined. Several proposals have been developed to have spaces on which L is quasi-compact. Such proposals are extremely effective when the map is smooth, see [B5] for a review, less so for discontinuous systems. Since in the case of expanding maps BV is very effective [Li13b], it is natural to investigate if one can construct a space, suitable for the study of invertible maps, that reduces to BV when the stable direction is absent.
In the next sections we define Banach spaces B 0,q and B 1,q that, when the stable direction is absent, reduce to L 1 and BV respectively (see Remark 2.11). Although we do not discuss discontinuous maps, this is certainly a first step to develop a viable alternative to the current approaches. To show that the space is potentially well behaved we prove the following Theorem.
(2) The peripheral spectrum of L consists of finitely many finite groups; in particular 1 is an eigenvalue. (3) Setting h * := Π 1 1, where Π 1 is the spectral projection of L associated with eigenvalue 1, the ergodic decomposition of h * corresponds to the spectral decomposition for the Anosov map and consists of the physical measures.
Observe that BV ⊂ B 1,q , see Remark 2.12, hence the above theorem implies that for Anosov maps the spectrum σ B 1,q (L) determines the decay of correlation for BV densities. In particular, for transitive Anosov maps the operator admits a spectral gap when acting on B 1,q , hence there exists ν q ∈ (0, 1) such that, for all ϕ ∈ C 1+q , rectifiable sets A and ν > ν q there exists a constant C: 5 Note that a similar estimate could be obtained using the spectral properties on spaces already existing in the literature and deducing the behaviour for BV densities by an approximation argument. However, this would produce a less sharp result (in particular a larger ν). In addition, the following are direct consequences of Theorem 2.1: • The Central Limit Theorem and other statistical properties for observables that 3 By det we mean the density of T * ω with respect to ω. 4 We still call such an extension L. 5 By rectifiable we mean that 1 A ∈ BV . are multipliers of BV via the usual spectral approach of analytic perturbation theory, e.g. see [G15].
• Statistical aspects of random perturbations: let T 0 be a transitive Anosov map. Let B T0 be a sufficiently small neighbourhood of T 0 in the C 1 -topology so that condition (2.2) is satisfied for all T ∈ B T0 with uniform constants. Let Ξ := sup l k ∂ l (DT −1 0 ) l,k C 1 and define the following family of maps One can study, for instance, iid compositions with respect to some product probability measure P defined on on G N Ξ . Spectral properties of the annealed transfer operator associated with the above random map follows from this work and stability results can be obtained using the current setting and the framework of [KL1].
Finally, given that BV ⊂ B 1,q , this space should be particularly useful to investigate numerically the spectrum of the transfer operator via Ulam-type perturbations, which proved to be very successful when dealing with expanding maps and BV functions [Liv]. Indeed previous investigations of Ulam approximation for Anosov systems left several questions unanswered due to the inadequacy of the Banach spaces used, e.g. [BKL] .

Foliations.
A fundamental ingredient in the understanding of hyperbolic maps is the study of dynamical foliations, hence a small digression is in order.
Definition 1. A C r t-dimensional foliation W is a collection {W α } α∈A , for some set A, such that the W α are pairwise disjoint, ∪ α∈A W α = M and for each ξ ∈ W α there exists a neighborhood B(ξ) such that the connected component of W α ∩ B(ξ) containing ξ, call it W (ξ), is a C r t-dimensional open submanifold of M . We will call F r the set of C r d s -dimensional foliations.
Definition 2. A foliation W is adapted to the cone filed C if, for each ξ ∈ M , T ξ W (ξ) ⊂ C(ξ). Let F r C be the set of C r d s -dimensional foliations adapted to C. Given a d s -foliation adapted to C we can associate to it local coordinates as follows. Let δ 0 > 0 be sufficiently small so that for each ξ ∈ M there exists a chart (V i , φ i ) with ξ ∈ V i and such that U i := φ i (V i ) contains the ball B δ0 (φ i (ξ)). 6 Also, In addition, we ask δ 0 to be small enough that the expression of DT in the above charts is roughly constant. See Lemma B.5 and its poof for the precise condition. 6 Here, and in the following, we use B δ (x) to designate {z ∈ R d ′ : x − z ≤ δ} for any d ′ ∈ N. 7 Refer to Definition 1 for the exact meaning of "connected component". Also note the abuse of notation since we use the same name for the sub-manifond in M and its image in the chart. 8 The fact that the intersection is non void and consists of exactly one point follows trivially from the fact that the foliation is adapted to the cone field, hence the two manifolds are transversal.
Remark 2.2. The above construction defines the triangular coordinates F ξ (x, y) = (F ξ (x, y), y) which describes locally the foliation. In fact, is a local chart of M in which the foliation is trivial (the leaves are all parallel). In the following we will often use such coordinates without mention if it will not create confusion. Also, to ease notation, we will confuse V i with φ i (V i ) when not ambiguous. In addition, we will use F to indicate the collection of maps {F ξ } and the same for F . Of course, F is not unique, since we can chose different charts for the same ξ, however different choices are equivalent so we assume that some choice has been made. Clearly F defines uniquely W .
Definition 3. For each r ∈ N and L > 0, let (2.6) Remark 2.3. Since the invariant foliation is not C r (in general it is only Hölder, although it consists of C r leaves) it does not belong to W r L for any L. Yet, it belongs to its closure, if L is large enough (see Remark B.3).
Remark 2.4. Note that the functions H F are related to the Jacobian of the stable holonomy (see Lemma B.7), hence it does not make sense to require them to be uniformly smooth. In general it is possible to control effectively only their Hölder norm, yet, restricted to the stable direction they turn out to be smooth. Indeed, this is the whole content of Appendix B.
Remark 2.5. The role of H F in the definition of W r D will become apparent in the proof of the Lasota-Yorke inequality in Proposition 3.2, namely in (3.7). Hence, controlling the sup ξ sup x∈U 0 , uniformly in n, is essential. Next we would like to define the evolution of a foliation W ∈ W r L under T . Let W n := T −n W := {T −n W α } α∈A . Clearly W n ∈ F r C , but much more is true.
Lemma 2.6. There exists n 0 ∈ N and L > 0 such that for all n ∈ N, n ≥ n 0 , L 1 ≥ L and W ∈ W r L1 , we have W n ∈ W r L1/2 .
Remark 2.7. By considering an appropriate power of the map, rather than the map itself, we can always reduce to the case n 0 = 1. We will do exactly this in the following.
Lemma 2.6 is proved in Appendix B. In fact we prove the more general Proposition B.1 which implies Lemma 2.6 (see Remark B.2).
2.5. Test Functions. Since we will want to be free to work with high order derivatives, it is convenient to choose a norm · C ρ equivalent to the standard one, for which C ρ is a Banach Algebra. We thus define the weighted norm in C ρ (M, M(m, n)), where M(m, n) is the set of the m × n (possibly complex valued) matrices, where, ̟ ≥ 2 is a parameter to be chosen later (see (3.10)), α is a multi The next Lemma is proven in Appendix A.
Remark 2.9. It is easy to verify that a different choice of the charts produces a uniformly equivalent class of norms.

A class of measures.
To be precise, we are going to define a Banach space of distributions. We will be interested in measures that belong to such a space. Define is defined in the natural manner using the norm (2.7) in the charts (V i , φ i ), also we use the charts to identify TxM with R d , hence Dψ ∈ M(d, d). 10 In fact, σ can be taken arbitrarily close to ν, but to keep the argument simple we will not insist on this.
The Lemma is proved in Appendix C. Note that Lemmata 2.6 and 2.10 imply T * Ω D,q,l ⊂ Ω D,q,l .
It is now time to define the norms. Given a function h ∈ C 1 (M, C) we define 11 (2.11) for any q ∈ N and some fixed a > 0 to be chosen later (see Proposition 3.2). We are then ready to define the Banach spaces. The space B 0,q is the Banach spaces obtained by completing C 1 (M, R) in the · 0,q norm. 12 We are not interested in making the same choice for the norm · − 1,q since this, in the case of d s = 0 would yield the Sobolev space W 1,1 rather than the space of function of bounded variations that we are interested in. We use thus the analogous of the standard procedure to define BV starting from W 1,1 . First let us define the new norm, for each h ∈ B 0,q , We then define B 1,q := {h ∈ B 0,q | h 1,q < ∞}. One can see Section 2.7 of [BKL] for a brief discussion of the general properties of such a construction.
Remark 2.11. If T is an expanding map, hence d s = 0, then the leaves are just points and ϕ W q = |ϕ| ∞ . The reader can easily check that B 0,q = L 1 and B 1,q = BV , as announced.
Remark 2.13. There is no problem in considering norms with higher smoothness, as in [GL]. We avoid it since it is not relevant for the issue we are presently exploring.
11 As already remarked the differential structure and the volume form are defined via the charts, thus, to be precise, Proof. Note that if ϕ ∈ C q , then there exists C q > 0 such that, for all W ∈ W r L , The other inclusion is proven similarly.

A Lasota-Yorke inequality
Our first goal is to show that L is bounded in the · 0,q , · 1,q norms, hence L extends uniquely to a bounded operator on B 0,q and B 1,q .
To prove our basic proposition (a Lasota-Yorke type inequality) we need first a small approximation Lemma.
Clearly ϕ ε ∈ C r , hence we only have to verify the other two properties. Note that Using the formula above and (3.1) we can estimate On the other hand, recalling (2.7), To verify the last inequality note that Proof. For each ε > 0 and (W, ϕ) ∈ Ω L,q,1 we define ϕ ε as in Lemma 3.1. Hence, Then, by Lemmata 2.10 and 3.1, Then, for each θ ∈ (σ, 1) there exists n 1 ∈ N and ε such that 2c −1 ̟ A 0 σ qn1 + C q,̟ B 0 ε ≤ θ n1 . Thus, taking the sup for (W, ϕ) ∈ Ω L,q,1 we have, for n ≤ n 1 Iterating yields for all n ≥ 1 Next, we prove the second part of the lemma. For each (W, ϕ) ∈ Ω L,q+1,d write It is then natural to decompose ϕ into an "unstable" and a "stable" part. More precisely consider the "almost unstable" foliation Γ = {γ s } s∈R ds made of the leaves, in some chart φ j , γ s = {(u, s)} u∈R du and its image T n Γ. The leaves of T n Γ can be expressed, in some chart φ i , in the form {(x,G(x, y)} for some functionG, smooth in the x variable, with ∂ xG ≤ 1 and the normalizationG(F (0, y), y) = y. On the other hand the leaves of W , in the same chart, have the form {(F (x, y), y)}. It is then natural to consider the change of variables (x, y) = Ψ(x ′ , y ′ ) where (x,G(x, y ′ )) = (F (x ′ , y), y). Writing ϕ = (ϕ 1 , ϕ 2 ), with ϕ 1 ∈ R du , ϕ 2 ∈ R ds we consider the decomposition Thus, recalling equation (3.3) and Lemma 2.10, ( 3.6) To estimate the above terms our first task is to compute the norm of div (ϕ s ), Accordingly, recalling (2.6), (3.7) Since, w W q+1 ≤ C n,̟ , recalling Definition 3 for all |α| ≤ q we have Thus, by Lemma 3.1 there exists ε > 0 (depending on ̟ and n) and g ε such that div (3.8) On the other hand, for each |α| = q + 1, using (3.4) and (3.5) we have 14 The value of the constant C * n is irrelevant, the point is that it depends only on T , the choice of coordinates and n.
where the last term bounds all the terms with at most q derivatives on v. Since the range of the matrix in the line above belongs to the image of the unstable cone under R, by (2.2) (and putting in the reminder all the terms with at most q derivatives of ϕ) we have Then Lemma 2.10 implies We can now chose n 2 ∈ N such that and finally we choose ̟ such that Accordingly, for all n ∈ {n 2 , . . . , 2n 2 }, We can then continue the estimate started in (3.6), recalling (3.8) we have: 15 Finally, choose a such that sup l≤2n2 C * l Aa −1 ≤ 1 2 , then taking the sup on ϕ, W we have, for all n ∈ {n 2 , . . . , 2n 2 }, Then, for each n ∈ N we can write n = kn 2 + m, m ≤ n 2 and iterating the above inequality we have, for all n ∈ N, Finally, if h ∈ B 1,q , then there exists {g k } ∈ B 0,q ∩ C 1 : g k B 0,q → h and g k − 1,q → h 1,q . Since, L n g k ∈ B 0,q ∩ C 1 and L n g k → L n h in B 0,q we have This finishes the proof of the second item in the proposition. The proof of the first item of the proposition follows from (3.2) and (3.10).

On the essential spectrum
In the previous section we have seen that L (or rather its extension that, with a slight abuse of notation, we still call L) belongs both to L(B 0,q , B 0,q ) and L(B 1,q , B 1,q ). Moreover Proposition 3.2 implies that the spectrum of L is contained in the unit disc. Next we want to study the essential spectrum (that is the complement of the point spectrum with finite multiplicity).
Let us prove the relative compactness of B − 1 . Since we can write we can assume, without loss of generality, that ϕ is supported in a given chart (V i , φ i ). Form now on we will work in such a chart without further mention. Let us define ϕ t to be the solution of the heat equation That is Then, for each small ε > 0, where ϕ ζ (x, y) := ϕ(x − ζ, y). Next, for each ζ ∈ R du we define the foliation F ζ (x, y) := (F (x − ζ, y) + ζ, y), note that the foliation W ζ defined by F ζ belongs to In addition, by (4.1) and integrating r times by parts Moreover, recalling (4.1), Definition 3 and Lemma B.7, which readily implies ϕ ε W r ≤ Cε −r . This, by [J], implies that |ϕ ε | C r ≤ Cε −r . Thus, recalling (4.2), we have, for each ε > 0, Since (C q+1 ) ′ embeds compactly in (C r ) ′ and Lemma 2.14 implies that B − 1 is a bounded subset of (C q+1 ) ′ it follows that B − 1 is relatively compact in (C r ) ′ . From this and equation (4.3) the relative compactness of B − 1 in B 0,q+1 readily follows. Hence the Lemma.

On the peripheral spectrum
The previous section implies, for each β ∈ (max{λ −1 , ν}, 1), the spectral decomposition is a finite rank operator, and the spectral radius of R is bounded by β. We choose β large enough so that |λ j | = 1. In this case, since the operator is power bounded, the Π j cannot contain Jordan blocks, thus Π j Π k = δ jk Π k . A simple computation based on (5.1) shows In other words h * is a measure. Then let h ∈ C 1 and ϕ ∈ C q , ϕ ≥ 0, Moreover, by a similar computation, It follows that φ j,l • T = e iϑj φ j,l , ω almost surely. On the other hand By the arbitrariness of g, h it follows e iϑj ψ j,l h * = Lψ j,l h * = ψ j,l • T −1 Lh * = ψ j,l • T −1 h * , which implies ψ j,l • T −1 = e iϑj ψ j,l , h * dω almost surely. Note that this implies that, for all k ∈ N, ψ k j,l •T −1 = e iϑj k ψ k j,l , thus L(ψ k j,l h * ) = ψ k j,l •T −1 Lh * = e iϑj k h * . By an approximation argument one can prove that ψ k j,l h * ∈ B 1,q . But then it follows that {e iϑjk } ⊂ σ(L) and since the operator is quasi-compact it can have only finitely many isolated eigenvalues: we must have ϑ j = 2πkj nj . In other words the peripheral spectrum of L must consist of finitely many finite groups.
It follows that there existsm ∈ N such that the peripheral spectrum of Lm consists of only the eigenvalue 1 with associated eigenprojector Π = N l=1 ψ l h * M hφ l where the ψ l ∈ {ψ j,i } and φ l ∈ {φ j,i }. Moreover, (5.3) implies Accordingly, the rest of the spectrum will be contained in a disk strictly smaller than one: that is Lm = Π + Q where Q n 1,q ≤ Cσ n for some C > 0 and σ ∈ (0, 1). In addition, note that (5.2) implies Π1 = h * .
A more precise result can be easily obtained.
Lemma 5.1. The ergodic decomposition of h * corresponds to the spectral decomposition for the Anosov map and consists of the physical measures.
On the other hand, since h is bounded, this readily implies By an obvious approximation argument the same can be proven for each h ∈ C 0 (M, R). This implies that the ergodic decomposition of h * consists of the physical measures. It is well known that these are the SRB measures of the system.
Remark 5.2. If the map is topologically transitive, then the physical measure is unique and so are the physical measures of the powers of the map. Hence the map is mixing, and no other eigenvalue of modulus one exist. Thus, the transfer operator has a spectral gap and the map is exponentially mixing for BV observables.

Appendix A. Norms estimates
We provide few tools on how to estimate C q norms of product and composition of functions. These are well known facts, yet it is not so easy to find in the literature the exact statements need here, so we provide them for the reader convenience.
Proof of Lemma 2.8. Let ϕ, ψ ∈ C ρ (M, C). First we prove, by induction on ρ, Indeed, it is trivial for ρ = 0 and from which (A.1) follows taking the sup on on α, i and since ρ k + ρ ρ+1−k = ρ+1 k . The first statement of the Lemma readily follows: The extension to functions with values in the matrices is trivial since we have chosen a norm in which the matrices form a normed algebra.
To prove the second inequality of the Lemma we proceed again by induction on ρ. The case ρ = 1 is trivial from the definition of the norm. Let us assume that the statement is true for every k ≤ ρ and show it for ρ + 1. By the definition of · C ρ , By hypothesis, 16 we have Finally notice that the term with q = 0 in the sum above is exactly the first term of the r.h.s. of (A.2), which gives the result for ρ + 1 and proves the induction.
Remark A.1. Notice that, for ϕ, ψ ∈ C ρ , the definition of the norm and Lemma 2.8 imply

Appendix B. Foliations: regularity properties
This appendix is devoted to proving Lemma 2.6 and other few technical Lemmas. In essence we study the behaviour of foliations under iteration. This is very similar to what is done in the construction of the invariant foliations and in the study of their regularity properties, including the regularity of the holonomies. The reason to redo it here without appealing to the literature is that we need these facts in an unconventional form. In particular, we could not find anywhere in the literature the infinitesimal characterisation of the holonomy used here: A characterization hopefully very helpful in the study of discontinuous hyperbolic maps.
Given such a new twist in the theory, we think it is appropriate to present a more general result: we will control also the regularity of the leaves, and of their tangent spaces, in the unstable direction although this is not needed in the present paper. More precisely we will see that the derivatives of the foliation along the leaves vary in a τ 0 -Hölder manner. The optimal τ 0 is well known to depend from a bunching condition [PSW, HW]. We ignore this issue since it largely exceeds our present purposes and to investigate it would entail a lengthier argument. Note that Lemma 2.6 is a special case of Proposition B.1 below when choosing τ = 0. Given ϕ : M → R we define, for some δ ⋆ > 0, where d(·, ·) is the Riemannian distance and δ ⋆ ∈ (0, 1). Note that ϕ · φ C τ ≤ ϕ C τ φ C τ , so C τ is a Banach algebra. The same holds for matrix valued functions.
Although the above norms are all equivalent, they depend on δ ⋆ . We will choose δ ⋆ in (B.11). Let T ∈ C r and define, for τ ∈ [0, 1), Note that, recalling the cone definition (2.3), the Definition 2 and the subsequent definition of F , it follows that, for W ∈ W r,τ L , the corresponding F must satisfy ∂ y F ≤ 1 and F (x, y) ≤ x + y .
Remark B.2. Note that for τ = 0 the conditions in W r,τ L reduce to a control on the sup norm of the derivatives ∂ α y F (·, y), that is on the C r norm of F (·, y), exactly as in the definition of W r L in Definition 3. The control stated in Proposition B.1 on ∂ α y F (·, y) is known, as for ∂ α y H F (·, y) we are not aware of this result anywhere in the literature.
Remark B.3. Note that for each W ∈ W r,τ L , τ > 0, the foliation T n W converges to the invariant foliation (since the contraction of the cone fields implies that, for all x ∈ M , D T n x T −n T T n x W (T n x) converges to the stable distribution E s ). 17 Accordingly, for each τ ′ < τ , by compactness, T n W has a convergent subsequence, hence it converges to the stable foliation and all the quantities in the definition of W r,τ L converge as well. It follows that the stable foliation have C r leaves with derivatives in y uniformly τ ′ Hölder in x. Analogously, also H F and its derivative converge. This implies that the invariant foliation has a Holonomy uniformly absolutely continuous (see Lemma B.7 for the definitions of the Holonomy and its J F ). Similar results hold also in the case τ = 0, but the argument is a bit more involved.
Proof of Proposition B.1. The first step in proving the Proposition is to determine, for each ξ ∈ M and n ∈ N, the functions F n T −n ξ associated to W n . 18 Note that it suffices to compute the norms in (B.2) in a special neighborhood of 17 Here TxV is the tangent space of the manifold V at the point x and W (x) is the fiber of the foliation passing through x. While E s (x) is the stable subspace in TxM . 18 Since the point ξ in the present argument is fixed once and for all, in the following we will often suppress the subscript ξ. We will also suppress the n dependence if no confusion arises.
While, for |α| > 0, Hence the sup on y and ξ ′ can be computed taking the sup of the quantity in the curly bracket (and the same for H F ).
Let (V i , φ i ), (V j , φ j ) be the charts associated to ξ and T −n ξ respectively and consider the map S = φ j • T −n • φ −1 i . By a simple translation we can assume, w.l.o.g., that φ i (ξ) = 0 and φ j (T −n ξ) = 0. From now on we use (x, y) for the coordinates name at φ i (ξ) and (u, s) for the coordinates name at φ j (T −n ξ). By a linear change of coordinates, that leaves {y = 0} and {s = 0} fixed, we can have ∂ y F (0, 0) = ∂ s F n (0, 0) = 0. Such a change of coordinates may affect the norms yielding some extra (uniformly bounded) constant in the estimates. We will ignore this to simplify notations since its effect is trivial. Also remember that, by construction, F (x, 0) = x, F n (u, 0) = u.
For each x, the manifold {(F (x, y), y)} y∈R ds intersects the manifold {(z, G(z))} R du in a unique point determined by the equation (F (x, y), y) = (z, G(z)) which is equivalent to L(z, x) := z − F (x, G(z)) = 0. Since L(0, 0) = 0 we apply the implicit function theorem and obtain a function Γ : 19 Since the implicit function theorem yields a uniform domain D(Γ), of Γ, we can take δ 0 small enough so that D(Γ) ⊃ U 0 u .
To continue we need some estimates on DS. But, before that, it is convenient to make some choices and definitions whose meaning will become clear later in the proof. Let τ 0 ∈ (0, 1) be such that (B.8) σ 1 := max{ν, λ −1 } · λ 8τ0 + < 1. Next, let σ 1 < σ < 1, fix C ⋆ > 0 to be chosen later (see equations (B.16), (B.20) and (B.25)) and let n ⋆ be the smallest integer such that Remark B.4. Up to now δ 0 was arbitrary provided we choose it small enough: the requirements are in Section 2.4 where we fix the charts and in footnote 19. In the following we will have also a condition in equation (B.12) to apply the implicit function theorem, and we will use δ 0 < 1/8 in equation (B.14). All such choices can be sumarized by the condition δ 0 ≤ δ 1 for some δ 1 ∈ (0, 1/8) depending only on T . However in the next Lemma we will have a requirement depending on n.
To study the zeroes of Ξ we apply the implicit function theorem. Since (u, 0, u, G(β(u))) ∂ y Ξ(u, 0, u, G(β(u))) = det E(β(u), G(β(u))) det 1 − DG(β(u))∂ y F (Υ(u)) = 0 where we have used (B.6) and DG∂ y F ≤ η. Thus there exists a uniform (in n) neighborhood of (u, G(β(u)) where the implicit function theorem can be applied. Thus, we can choose δ 0 small enough so that, for each n ∈ N, there exist F n , Φ ∈ C r : Note that (B.14) Differentiating (B.13) with respect to s we obtain We now study ∂ α s F n when |α| ≥ 2. Differentiating (B.15) and setting ∆ where Θ 1 is a rational function of its arguments and, for an arbitrary matrix R, Differentiating further (B.17) we can prove, by induction, that, for all l ≤ r, where the Θ l are sums of functions k j -multilinear in ∂ j y F , for j ∈ {2, . . . , l}, such that l j=2 k j (j − 1) ≤ l. Indeed, we have seen that this is true for l = 2. On the other hand, if it is true for l−1, then differentiating (B.19) we produce several terms. Let us analyse them one by one. The term proportional to H, when differentiated with respect to ∂ l y F , yields the correct term proportional of H. When differentiated with respect to D k S • F yields a function of D k+1 S • F multiplied by ∂ y F · ∂ s Φ so the multilinearity with respect to ∂ j y F , for j ∈ {2, . . . , l}, is unchanged. When differentiating with respect to ∂ y F , the term gets multiplied by ∂ 2 y F . 20 Thus, calling k ′ j the multilinearities of the term obtained we have k ′ l = 1, k ′ 2 = 1 and all the other k ′ j are zero. That is l j=2 k ′ j (j − 1) = l − 1 + 1 = l. Next we must differentiate Θ l−1 . Again the only change in the multilinearity occurs when differentiating with respect to a ∂ m y F , m ∈ {1, . . . , l − 1}. If j = 2 then we have (calling again k ′ the new multilinearities) k ′ 2 = k 2 +1 and k ′ j = k j for j > 2, that is  ∂ l s F n ≤ [L 1 /2] (l−1) 2 . Next we estimate the Hölder norms of ∂ α s F n for |α| ≤ r − 1. We first treat the case |α| = 0. By strict cone field invariance and the continuity of the cone field it follows that, for all s, Analogously, by (B.13) S −1 F n (u, s) = (F • Ω(u, s), Φ(u, s)). Hence where we have used ∂ y F ≤ 1. Accordingly Next we prove an auxiliary lemma, which will be used repeatedly in the following.
Let us verify it: equation (B.36) shows that it is true for l = 0. Let us assume it true for l − 1, then differentiating the first term we obtain the correct term linear in ∂ l H F , the other terms are linear in ∂ l−1 H F and linear in ∂ 2 F • Ω (see equation (B.15)) hence p ′ = l − 1, k l−1,2 = 1 and all the other degree are zero, so p ′ + k l−1,2 ≤ l. Differentiating Θ l−1 with respect to D m S • F • Ω does not change the multilinearity indexes. Differentiating with respect to ∂ j s F n yields, for each p, a term with p ′ = p, q ′ p ′ ,j = q p,j − 1 multilinear in ∂ j s F n and q ′ p ′ ,j+1 = q p,j+1 + 1 multilinear in ∂ j+1 s F n . Thus p ′ + l+2 j=2 (k ′ p ′ ,j +q ′ p ′ ,j )(j −1) ≤ l. The same happens if one differentiates with respect to ∂ j y F •Ω for j ≥ 2. On the other hand differentiating with respect to ∂ y F • Ω yields a term in which p ′ = p, k p ′ ,2 ′ = k p,2 + 1, thus p ′ + l+2 j=2 (k ′ p ′ ,j + q ′ p ′ ,j )(j − 1) ≤ l. Finally, if we differentiate with respect to ∂ j y H F • Ω for 0 ≤ j < l − 1 we have a term with p ′ = p + 1 and k ′ p ′ ,j = k p,j , q ′ p ′ ,j = q p,j , thus, again p ′ + l+2 j=2 (k ′ p ′ ,j + q ′ p ′ ,j )(j − 1) ≤ l. Which proves the claim.  The claim follows then by induction and using the known structure of Θ l . 23 We use the convention that q p,l+2 = 0 and ∂ −1 s H F n = 1.
We conclude the section by clarifying the relation between the function H F and the holonomy associated with the foliation F. The next Lemma shows that the Jacobian of the Holonomy can be seen as a flow of which H F is the "generator".
Lemma B.7. If W ∈ W r,0 L , then there exists C > 0 and r 0 > 0 such that, for each ξ ∈ M , 0 < r < r 0 and (x ′ , y ′ ) ≤ r, we have det(∂ x F ξ )(x ′ , ·) C q ≤ C. More precisely, setting J F ξ (x, y) = det(∂ x F ξ )(x, y), we have Proof. Let (x ′ , y ′ ) as in the Lemma's assumption. First of all note that for each vector e i ∈ R d we have
Remark B.8. Lemma B.7 implies that, for each measurable set B ⊂ R du and |β| ≤ r − 1, holds Note that the above inequality does not involve ∂ x F , hence it may hold also for non differentiable F . The same Remark holds also for equation (B.40). In other words if we consider the true invariant foliation, where ∂ x F may make no sense, still H F is well defined (see Remark B.3 for details), and so, by equation (B.40), is the Jacobian of the Holonomy J F .
From this and recalling the definition (2.7) follows ϕ • T n T −n W q ≤ A 0 ϕ W q . On the other hand, recalling (2.8), there exists B 0 > 0 such that