Sharp large deviations for hyperbolic flows

Abstract

For hyperbolic flows ${\phi }_t$ we examine the Gibbs measure of points $w$ for which $${\int }_0^{T}G({\phi }_{t}w)dt-aT\in (-e^{-\epsilon n},e^{-\epsilon n})$$ as $n\to \infty$ and $T\ge n$, provided $\epsilon >0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(-e^{-\epsilon n},e^{-\epsilon n})$ is exponentially shrinking as $n\to \infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a){\epsilon }_n{e}^{\gamma (a)T}$ and rate function $\gamma (a)\le 0.$ 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 $g_n(t)$ having an asymptotic as $n\to \infty$ and $t\ge n.$

Annales Henri Poincare, to appear

References in corpus (2)

• On mixing and the local central limit theorem for hyperbolic flows
• Spectral estimates for Ruelle operators with two parameters and sharp large deviations