Sharp large deviations for hyperbolic flows

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