Distribution of periods of closed trajectories in exponentially shrinking intervals

Abstract

We examine the asymptotics of the number of the closed trajectories $\gamma$ of hyperbolic flows ${\varphi }_t$ whose primitive periods $T_{\gamma }$ lie in exponentially shrinking intervals $(x-e^{-\delta x},x+e^{-\delta x}),\delta >0,x\to +\infty .$ 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.

Cited by in corpus (2)

• Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows
• Dolgopyat type estimates for pinched open billiard flows (Revised)