Sharp large deviations for hyperbolic flows

For hyperbolic flows $varphi_t$ we examine the Gibbs measure of points $w$ for which $$int_0^T G(varphi_t w) dt - a T in (- e^{-epsilon n}, e^{- epsilon n})$$ as $n to infty$ and $T geq n$, 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 $gamma(a) leq 0.$ 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 infty$ and $t geq n.$