Sharp Large Deviations for Hyperbolic Flows

For hyperbolic flows φ_t, we examine the Gibbs measure of points w for which ∫0TG(φtw)dt-aT∈(-e-ϵn,e-ϵn)as n→ ∞ and T≥ 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→ ∞ leads to several difficulties. Under some geometric assumptions, we establish a sharp large deviation result with leading term C(a) ϵ_ne^γ(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 g_n(t) having an asymptotic as n→ ∞ and t≥ n.