Sharp large deviations for some hyperbolic systems

Abstract

We prove a sharp large deviation principle concerning intervals shrinking with sub-exponential speed for certain models involving the Poincar\'e map related to a Markov family for an Axiom A flow restricted to a basic set $\Lambda$ satisfying some additional regularity assumptions.Comment: arXiv admin note: substantial text overlap with arXiv:0810.112

Spectral estimates for Ruelle operators with two parameters and sharp large deviations

“…

We obtain spectral estimates for the iterations of Ruelle operators L f +(a+ib)τ +(c+id)g with two complex parameters and Hölder continuous functions f, g generalizing the case Pr(f ) = 0 studied in [9]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [8].

…”

“…Our second problem concerns the analysis of the asymptotic of µ{x : g n (x) − τ n (x)p ∈ (−δ n , δ n )}, n → ∞, and for p = R τ Gdm F we obtain a large deviation result. On the other hand, as in the previous paper [8], we examine the measure of points x ∈ R for which the difference g n (x) τ n (x) − p stays in an exponentially shrinking interval. Next, assume that the flow σ t τ is topologically weak mixing and the function G is not cohomologous to a constant function.…”

Ruelle transfer operators with two complex parameters and applications

“…4, 5, we study also the analytic continuation of ζ(s, iw) for P f − η 0 < Re s and w ∈ R, |w| ≥ η 0 , in the case when F, G : Λ −→ R are Lipschitz functions (see Theorem 7). This analytic continuation combined with the arguments in [22] opens some new perspectives for the investigation of sharp large deviations for Anosov flows with exponentially shrinking intervals in the spirit of [12].…”

A uniform estimate for rate functions in large deviations

“…For some hyperbolic systems, large deviation principles similar to (1), however with shrinking intervals, have been established recently in [8,9].…”