Distribution of periods of closed trajectories in exponentially shrinking intervals

Details Distribution of periods of closed trajectories in exponentially shrinking intervals Distribution of periods of closed trajectories in exponentially shrinking intervals

2010-08-25

arxiv.org/abs/1008.4308v5

Mathematics

Dynamical Systems

Scientific paper

We examine the asymptotics of the number of the closed trajectories $\gamma$ of hyperbolic flows $\phi_t$ 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.

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