Sharp Large Deviations for Hyperbolic Flows
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Annales Henri Poincar´ e
Sharp Large Deviations for Hyperbolic Flows Vesselin Petkov
and Luchezar Stoyanov
Abstract. For hyperbolic flows ϕt , we examine the Gibbs measure of points w for which T G(ϕt w)dt − aT ∈ (−e−n , e−n ) 0
as n → ∞ and T ≥ n, provided > 0 is sufficiently small. This is similar to local central limit theorems. The fact that the interval (−e−n , e−n ) is exponentially shrinking as n → ∞ leads to several difficulties. Under some geometric assumptions, we establish a sharp large deviation result with leading term C(a)n eγ(a)T and rate function γ(a) ≤ 0. The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions gn (t) having an asymptotic as n → ∞ and t ≥ n. Mathematics Subject Classification. Primary 37D35, 37D40, Secondary 60F10.
1. Introduction Let ϕt : M −→ M be a C 2 weak mixing Axiom A flow on a compact Riemannian manifold M , and let Λ be a basic set for ϕt . The restriction of the flow on Λ is a hyperbolic flow [11]. For any x ∈ M let Ws (x), Wu (x) be the local stable and unstable manifolds through x, respectively (see [2,7,11]). It follows from the hyperbolicity of Λ that if 0 > 0 is sufficiently small, there exists 1 > 0 such that if x, y ∈ Λ and d(x, y) < 1 , then Ws0 (x) and ϕ[−0 ,0 ] (Wu0 (y)) intersect at exactly one point [x, y] ∈ Λ (cf. [7]). That is, there exists a unique t ∈ [−0 , 0 ] such that ϕt ([x, y]) ∈ Wu0 (y). Setting Δ(x, y) = t, defines the so-called temporal distance function. Here and throughout the whole paper we denote by d(·, ·) the distance on M determined by the Riemannian metric. Let R = {Ri }ki=1 be a fixed (pseudo) Markov family of pseudo-rectangles Ri = [Ui , Si ] = {[x, y] : x ∈ Ui , y ∈ Si } (see Sect. 2). Set R = ∪ki=1 Ri , U =
V. Petkov and L. Stoyanov
Ann. Henri Poincar´e
∪ki=1 Ui . Consider the Poincar´e map P : R −→ R, defined by P(x) = ϕτ (x) (x) ∈ R, where τ (x) > 0 is the smallest positive time with ϕτ (x) (x) ∈ R (first return time function). The shift map σ : R −→ U is given by σ = πU ◦ P, where πU : R −→ U is the projection along stable leaves. Define a (k × k) matrix A = {A(i, j)}ki,j=1 by 1 if P(Int Ri ) ∩ Int Rj = ∅, A(i, j) = 0 otherwise. Following [2], it is possible to construct a Markov family R so that A is irreducible and aperiodic. Consider the suspension space Rτ = {(x, t) ∈ R × R : 0 ≤ t ≤ τ (x)}/ ∼, where by ∼ we identify the points (x, τ (x)) and (Px, 0). The suspension flow on Rτ is defined by ϕτt (x, s) = (x, s + t) taking into account the identification ∼ . For a H¨ older continuous function f on R, the topological pressure PrP (f ) with respect to P is defined by PrP (f ) = sup h(P, m) + f dm , m∈MP
where MP denotes the space of all P-invariant Borel probability measures and h(P, m) is the entropy of P with respect to m. We say that u and v are cohomologous, and we denote this by u ∼ v if there exists a continuous function w such that u = v + w ◦ P − w. The flow ϕt on Λ is naturally related to th
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