On the distribution of periodic orbits

TL;DR

This paper investigates the distribution of periodic orbits in non-invertible, non-uniformly hyperbolic dynamical systems, extending classical results to broader contexts and providing new insights into pressure, measures, and deviations.

Contribution

It extends Bowen, Ruelle, and others' results to non-invertible, non-uniformly hyperbolic maps, showing how periodic orbits approximate invariant measures and compute topological pressure.

Findings

Topological pressure can be computed from expanding periodic orbits.

Hyperbolic ergodic measures are approximated by expanding periodic orbits.

Certain equilibrium states are identified as Bowen measures.

Abstract

Let $f:M\to M$ be a $C^{1+\epsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic orbits of $f|\Lambda$. These results are non-invertible and, in particular, non-uniformly hyperbolic versions of well-known results by Bowen, Ruelle, and others in the case of hyperbolic diffeomorphisms. We show that the topological pressure $P_{\rm top}(\varphi)$ can be computed by the values of the potential $\varphi$ on the expanding periodic orbits and also that every hyperbolic ergodic invariant measure is well-approximated by expanding periodic orbits. Moreover, we prove that certain equilibrium states are Bowen measures. Finally, we derive a large deviation result for the periodic orbits whose time averages are apart from the space average of a given hyperbolic invariant measure.