Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows

For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls $B^s(x,\ep)$ on local stable manifolds at Lyapunov regular points $x$. We prove that this decay rate is similar for all sufficiently small values of $\epsilon>0$. From this and the main result in \cite{kn:St1}, we derive strong spectral estimates for Ruelle transfer operators for contact Anosov flows with Lipschitz local stable holonomy maps. These apply in particular to geodesic flows on compact locally symmetric manifolds of strictly negative curvature. As is now well known, such spectral estimates have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, asymptotics of closed orbit counting functions, etc.


Introduction
Consider a non-linear system of differential equations of the forṁ (1.1) where f : U × R −→ R n is a continuously differentiable map for some open ball U with center 0 in R n and f (t, 0) = 0 for all t ∈ R. Assuming that the null solution of (1.1) is asymptotically stable (see e.g. [CL]), one defines a semi-flow ϕ t : U × [0, ∞) −→ U such that ϕ t (z, s) = x(t + s), where x is the solution of (1.1) with x(s) = z. One may then ask the question whether for all sufficiently small 0 < δ 1 < δ 2 there exists a constant C > 0 depending only on δ 1 and δ 2 (and f ) such that diam(ϕ t (B(0, δ 2 ))) ≤ Cdiam(ϕ t (B(0, δ 1 ))) for all t ≥ 0, where B(0, δ) denotes the (closed) ball with center 0 and radius δ in R n . We do not know what happens in the general case, however it follows from the arguments in the present paper that under a certain (Lyapunov regularity) condition at 0, the answer to the above question is affirmative. In fact, we consider a more complicated situation. Let φ t : M −→ M be a C 2 Anosov flow on a C 2 compact Riemann manifold M . For any x ∈ M and a sufficiently small δ > 0 consider the closed δ-ball B s (x, δ) = {y ∈ W s ǫ (x) : d(x, y) ≤ δ} on the local stable manifold W s ǫ (x). For any y ∈ W s ǫ (x) we know that d(φ t (x), φ t (y)) → 0 exponentially fast as t → ∞. Moreover, we have uniform estimates for the exponential rate of convergence, so for any x ∈ M and any given δ > 0, diam(φ t (B s (x, δ))) → 0 exponentially fast as t → ∞. However, in general it is not clear whether for any constants 0 < δ 1 < δ 2 the ratio diam(φ t (B s (x, δ 2 ))) diam(φ t (B s (x, δ 1 ))) is uniformly bounded for t > 0 and x ∈ M (although a similar property is obviously satisfied by the linearized flow dφ t , considering balls on corresponding tangent planes). It appears that in general this problem is rather subtle, and it is not clear at all whether one should expect a positive solution without any extra assumptions.
Here we consider a similar problem on the set L of Lyapunov regular points in M -see section 3.1 for the terminology. We prove the following.
A similar result can be proved for non-uniformly hyperbolic flows. The above has an important consequence concerning cylinders in a symbolic coding of the flow defined by means of a Markov family -see Theorem 4.2 below for details.
The motivation for this work came from [St1], where assuming the properties (a) and (b) in Theorem 4.2 below, Lipschitzness of the local stable holonomy maps and a certain non-integrability condition we prove strong spectral estimates for arbitrary potentials over basic sets for Axiom A flows, similar to those established by Dolgopyat [D] for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on compact manifolds with C 1 jointly nonintegrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). It is known that such strong spectral estimates lead to deep results in a variety of areas which are difficult (if not impossible) to obtain by other means (see e.g. [PoS1], [PoS2], [PoS3], [An], [PeS1] [PeS2], [PeS3]). Let for some ǫ > 0 and z i ∈ M (cf. section 2 for details). The first return time function τ : is the projection along the leaves of local stable manifolds, provides a natural symbolic coding of the flow. To avoid dealing with boundary points in U , consider the set U of all u ∈ U whose orbits do not have common points with the boundary of R (see section 2).
Given a Lipschitz real-valued function f on U , set g = g f = f − P τ , where P = P f ∈ R is the unique number such that the topological pressure Pr σ (g) of g with respect to σ is zero (cf. e.g. [PP]). For a, b ∈ R, one defines the Ruelle transfer operator L g−(a+ib)τ : C Lip ( U ) −→ C Lip ( U ) in the usual way (cf. section 2). Here C Lip ( U ) is the space of Lipschitz functions g : U −→ C. By Lip(g) we denote the Lipschitz constant of g and by g 0 the standard sup norm of g on U .
We will say that the Ruelle transfer operators related to the function f on U are eventually contracting if for every ǫ > 0 there exist constants 0 < ρ < 1, a 0 > 0 and C > 0 such that if a, b ∈ R satisfy |a| ≤ a 0 and |b| ≥ 1/a 0 , then for every integer m > 0 and every This implies in particular that the spectral radius of L f −(P f +a+ib)τ on C Lip ( U ) does not exceed ρ.
From Theorem 1.1 (or rather its consequence -Theorem 4.2 below) and the main result in [St1] we derive the following.
Theorem 1.2. Let φ t : M −→ M be a C 2 transitive contact Anosov flow on a C 2 compact Riemann manifold with uniformly Lipschitz local stable holonomy maps. Then for any Lipschitz real-valued function f on U the Ruelle transfer operators related to f are eventually contracting.
The reader is referred to section 2 below for the definition of local holonomy maps. In general these are only Hölder continuous. It is known that uniform Lipschitzness of the local stable holonomy maps can be derived from certain bunching condition concerning the rates of expansion/contraction of the flow along local unstable/stable manifolds over M (see [Ha], [PSW]).
A result similar to Theorem 1.2 is true for general (non necessarily contact) Anosov flows, however one has to assume in addition a local non-integrability condition (see condition (LNIC) in [St1]). Using a smoothing procedure as in [D], an estimate similar to that in Theorem 1.2 holds for the Ruelle operator acting on the space F γ (U ) of Hölder continuous functions with respect to an appropriate norm.
For geodesic flows on locally symmetric spaces of negative curvature it is well known that the local stable and unstable manifolds are smooth (C ∞ ), so the corresponding local holonomy maps are smooth as well. Thus, as an immediate consequence of Theorem 1.2 one obtains the following. As mentioned above, there are various consequences that can be derived from results like Theorem 1.2 (or Theorem 1.3). Here we state one of these.
As in [St1], one can use Theorem 1.2 and an argument of Pollicott and Sharp [PoS1] to get certain information about the Ruelle zeta function where γ runs over the set of primitive closed orbits of φ t and ℓ(γ) is the least period of γ. Let h T denote the topological entropy of φ t .
In fact, a direct application of Theorem 5 in [PeS3] gives a more precise estimate of the number of closed trajectories of the flow with primitive periods lying in exponentially shrinking intervals -we refer the reader to section 6 in [PeS3] for details.
Section 2 contains some basic definitions and preliminary facts. In section 3 we compare diameters of balls with respect to Bowen's metric on unstable manifolds and prove the analogue of Theorem 1.1 for unstable manifolds. From this Theorem 1.1 is derived easily. Finally in section 4 we consider cylinders in the set U defined by means of a Markov family, and prove two properties of the decay rates of the diameters of such cylinders (Theorem 4.2), assuming that the local stable holonomy maps are uniformly Lispchitz. We do not know whether the same properties hold for any Anosov flow. Theorem 1.2 is then derived using Theorem 4.2 and the argument in section 5 of [St1].

Preliminaries
Throughout this paper M denotes a C 2 compact Riemann manifold, and φ t : M −→ M (t ∈ R) a C 2 flow on M . The flow is called hyperbolic if M contains no fixed points and there exist constants C > 0 and 0 < λ < 1 such that there exists a dφ t -invariant decomposition T is the one-dimensional subspace determined by the direction of the flow at x, dφ t (u) ≤ C λ t u for all u ∈ E s (x) and t ≥ 0, and dφ t (u) ≤ C λ −t u for all u ∈ E u (x) and t ≤ 0. The flow φ t is called an Anosov flow on M if the periodic points are dense in M (see e.g. [KH]). The flow is called transitive if it has a dense orbit, and contact if there exists a C 2 flow invariant one form ω on M such that ω ∧ (dω) n is nowhere zero, where dim(M ) = 2n + 1.
For x ∈ M and a sufficiently small ǫ > 0 let . Admissible subsets of W s ǫ (z) are defined similarly. As in [D], a subset R of Λ will be called a rectangle if it has the form R = [U, S] = {[x, y] : x ∈ U, y ∈ S}, where U and S are admissible subsets of W u ǫ (z) and W s ǫ (z), respectively, for some z ∈ M . In what follows we will denote by Int u (U ) the interior of U in the set W u ǫ (z). In a similar way we define Int s (S), and then set Int The interiors of these sets in the corresponding leaves are defined by Since the set L of Lyapunov regular points (see section 3.1 below) is dense, without loss of generality we will assume that z The existence of a Markov family R of an arbitrarily small size χ > 0 for φ t follows from the construction of Bowen [B] (cf. also Ratner [Ra]).
From now on we will assume that R = {R i } k i=1 is a fixed Markov family for φ t of small size

For any integer m ≥ 1 and any function
It is well-known (see [B]) that U is a residual subset of U and has full measure with respect to any Gibbs measure on U . Clearly in general τ is not continuous on U , however τ is essentially Lipschitz on U in the sense that there exists a constant L > 0 such that if The hyperbolicity of the flow on M and the additional assumption (in section 4 below) that the local stable holonomy maps are uniformly Lipschitz implies the existence of constants c 0 ∈ (0, 1] and γ 1 > γ > 1 such that whenever σ j (u 1 ) and σ j (u 2 ) belong to the same U i j for all j = 0, 1 . . . , m.
3 Comparison of ball diameters
Fix for a moment µ > 0 with the above properties. Then, for x ∈ L we have an f -invariant decomposition into subspaces of constant dimensions n 1 , . . . , n s such that for some Lyapunov µ-regularity function R = R µ : L −→ (1, ∞), i.e. a function with 1 (x) and u (2) ∈ E u 2 (x). We will denote by · the norm on E u (x) generated by the Riemann metric, and we will also use the norm |u| = max{ u (1) , u (2) }. Taking the regularity function R(x) appropriately (see [P1], [BP] or [PS]), we may assume that It follows from the general theory of partial hyperbolicity (see [P1], [P2], [BP]) that the invariant bundle { E u 2 (x)} x∈L is uniquely integrable over L, i.e. there exists a continuous finvariant family {W u,2 q(x) (x)} x∈L of C 2 submanifolds W u,2 q(x) (x) of M tangent to the bundle E u 2 for some Lyapunov µ/2-regularity functionq =q µ/2 : L −→ (0, 1). Moreover, it follows from Theorem 6.6 in [PS] and (3.1) that there exists an f -invariant family {W u,1 q(x) (x)} x∈L of C 1+α submanifolds W u,1 q(x) (x) of M tangent to the bundle E u 1 . (However this family is not unique in general.) For each x ∈ L fix an f -invariant family {W u,1 q(x) (x)} x∈L with the latter properties. Then we can find a Lyapunov µ-regularity function q = q µ : L −→ (0, 1) and for any x ∈ L a C 1+α diffeomorphism (3.7) We will assume without loss of generality that the regularity function R satisfies For any x ∈ L consider the C 1+α map (defined locally near 0) Given y ∈ L and any integer k ≥ 1 we will use the notation at any point where these sequences of maps are well-defined. It is well known (see e.g. the Appendix in [LY] or section 3 in [PS]) that there exist Lyapunov µ-regularity functions Γ = Γ µ : L −→ [1, ∞) and q = q µ : L −→ (0, 1) and for each x ∈ L a norm and for any x ∈ L and any integer n ≥ 0, assumingf j , q(f j (x))) are well-defined for all j = 1, . . . , n, the following hold: and (3.14) Clearly each of the above inequalities provides a corresponding inequality involving the norm · . For example (3.9) and (3.10) imply for all x ∈ L and all u, v ∈ E u 2 (x).
For a non-empty set X ⊂ E u (x) set ℓ(X) = sup{ u : u ∈ X} .
Given z ∈ L and p ≥ 1, setting x = f p (z), define Notice that u ∈ E u 1 (z) impliesf p z (u) ∈ E u 1 (f p (z)) wheneverf p z (u) is well-defined. Theorem 3.1 will be derived from Lemma 3.3 below and the following proposition.
Proposition 3.2. There exists a 12µ/α-regularity function ω : L −→ (0, 1) with ω(x) ≤ q(x) for all x ∈ L and a 4µ-regularity function G : L −→ [1, ∞) such that for any x ∈ L, any δ ∈ (0, ω(x)] and any integer p ≥ 1 for z = f −p (x) we have The proof of Proposition 3.2 takes most of this section. Taylor's formula (see also section 3 in [PS]) implies that there exists a Lyapunov µ-regularity Fix for a moment x ∈ L and an integer p ≥ 1, set z = f −p (x) and given v ∈ E u (z; q(z)), set for any j = 0, 1, . . . , p (assuming that these points are well-defined).

Corollary 3.4. Under the assumptions of Lemma 3.3, for
Proof. We just apply Lemma 3.3 replacing v by u. Sincef p z (u) ∈ E u 1 (x), we have u p = u (1) p .
Proof of Theorem 1.1. Consider the Anosov flow ψ t = φ −t on M . Clearly this flow has the same set L of Lyapunov regular points. Let ω and G be Lyapunov regular functions satisfying the requirements of Theorem 3.1 for the flow ψ t . We will denote by W s δ (x) and W u δ (x) the local stable and unstable manifolds for the flow ψ t . Clearly W s δ (x) = W u δ (x) and W u δ (x) = W s δ (x). Given x ∈ L, t > 0 and 0 < δ 1 < δ 2 ≤ ω(x), set x ′ = φ t (x), and notice that φ t (B s Using Theorem 3.1 for the flow ψ, it follows that there exists a constant K(δ 1 , δ 2 ) ≥ 1 such that diam (B u t 4 Decay of cylinder diameters in a Markov coding be a fixed Markov family as in section 2. Define the matrix A = (A ij ) k i,j=1 by A ij = 1 if P(Int(R i )) ∩ Int(R j ) = ∅ and A ij = 0 otherwise. According to [BR] (see section 2 there), we may assume that R is chosen in such a way that A M 0 > 0 (all entries of the M 0 -fold product of A by itself are positive) for some integer M 0 > 0. In what follows we assume that the matrix A has this property. Given a finite string ı = (i 0 , i 1 , . . . , i m ) of integers i j ∈ {1, . . . , k}, we will say that ı is admissible if for any j = 0, 1, . . . , m − 1 we have A i j i j+1 = 1. Given an admissible string ı, denote by • C [ı] the set of those x ∈ U so that σ j (x) ∈ Int u (U i j ) for all j = 0, 1, . . . , m. The set In what follows the cylinders considered are always defined by finite admissible strings. Given x ∈ U i for some i and r > 0 we will denote by B U (x, r) the set of all y ∈ U i with d(x, y) < r.
It is easy to see that diam(C[ı]) → 0 exponentially fast as m → ∞. A much more subtle question is if there exists a constant ρ ∈ (0, 1) such that for any cylinder C = C[i 0 , i 1 , . . . , i m ] and any subcylinder C ′ = C[i 0 , i 1 , . . . , i m , i m+1 ] we have diam(C ′ ) ≥ ρ diam(C). Using Theorem 1.1 here we show that this is always the case under some regularity assumptions about the flow.
The following is an easy consequence of (2.1). In what follows we will assume that ρ 1 ∈ (0, 1) and C 1 > 0 are fixed constants with the above property. Fix a constant ǫ > 0 such that From now on we will assume that the local stable holonomy maps through Λ are uniformly Lipschitz. Then there exists a constant L ≥ 1 such that d(π y (z), π y (z ′ )) ≤ L d(z, z ′ ) for all x, y ∈ M with d(x, y) < ǫ 1 and z, z ′ ∈ W u ǫ 1 (x). (See section 2 for the choice of ǫ 1 .) Given i = 1, . . . , k, according to the choice of the Markov family {R i }, the projection is also Lipschitz. Thus, we may assume the constant L ≥ 1 is chosen sufficiently large so that d(ψ i (u), ψ i (v)) ≤ L d(u, v) for all u, v ∈ W i and all i = 1, . . . , k.
Next, if V = W u R (x) is the unstable leaf of some point x ∈ R i and ı = (i 0 = i, i 1 , . . . , i m ) is an admissible sequence, consider the generalized cylinder for any choice of V and the admissible sequence ı. For V as above, x ∈ V and δ > 0 set Proof of Theorem 4.2. Notice that for any (admissible) ı we have σ m ( C[i 0 , . . . , i m ]) = U im .
By Theorem 3.1, there exists a constant K = K(δ 1 , δ 2 ) > 0 (depending on δ 1 and δ 2 which are constants in our case) such that diam (B u t However, using the above information about C and C ′ , as in the proof of Proposition 3.3 in [St1], one easily observes that C ′ ⊃ B u t (z, δ 1 ) and C ⊂ B u t (z, δ 2 ). Thus, diam(C ′ ) ≥ 1 K G 0 diam(C). Combining the latter with (4.3) gives diam(C[ı ′ ]) ≥ 1 L 2 K G 0 diam(C[ı]). This proves part (a) for m > p. Since there are only finitely many cylinders of length ≤ p, it follows immediately that there exists ρ ∈ (0, 1/(L 2 KG 0 )] which satisfies the requirements of part (a).
(b) This follows easily combining a simple modification of the proof of Proposition 3.3(b) in [St1] with an argument similar to the above. We omit the details.
Proof of Theorem 1.2. This now follows from the main result (Theorem 1.1) in [St1], or rather from the proof of this theorem in section 5 in [St1]. What the latter assumes is a local non-integrability condition (LNIC), uniformly Lispchitz local stable holonomy maps and the so called (see section 1 in [St1]) regular distortion along unstable manifolds. In our case the flow is contact, so the condition (LNIC) follows from Proposition 6.2 in [St1]. What concerns regular distortion along unstable manifolds, one should note that section 5 in [St1] is only using a consequence of this property, namely the properties of cylinders described in Proposition 3.3 in [St1]. These properties are exactly the properties (a) and (b) in Theorem 4.2 above. Thus, under the assumptions of Theorem 1.2 above the argument from section 5 in [St1] applies and proves that the Ruelle transfer operators related to f are eventually contracting for any Lipschitz real-valued function f on U .