Distribution of periods of closed trajectories in exponentially shrinking intervals

Abstract

We examine the asymptotics of the number of the closed trajectories $γ$ of hyperbolic flows $ϕ_t$ whose primitive periods $T_γ$ lie in exponentially shrinking intervals $(x - e^{-δx}, x + e^{-δx}),\:δ> 0,\: x \to + \infty.$ Our results holds for hyperbolic dynamical systems having a symbolic model with a non-lattice roof function $f$ under the assumption that the corresponding Ruelle operator related to $f$ satisfies strong spectral estimates. In particular, our analysis works for open billiard systems and for the geodesics flow on manifolds with constant negative curvature.