On the distribution of periodic orbits

Pages: 949 - 966, Volume 26, Issue 3, March 2010

doi:10.3934/dcds.2010.26.949       Abstract        Full Text (251.6K)

Katrin Gelfert - IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil (email)
Christian Wolf - Department of Mathematics, Wichita State University, Wichita, Kansas, 67260, United States (email)

Abstract: Let f : $M\to M$ be a $C^{1+\varepsilon}$-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_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.

Keywords:  Pesin theory, non-uniformly hyperbolic dynamics, topological pressure, equilibrium states, large deviation.
Mathematics Subject Classification:  Primary: 37D40; Secondary: 37D50, 37C35.

Received: January 2009;      Revised: September 2009;      Published: December 2009.