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September  2010, 26(3): 949-966. doi: 10.3934/dcds.2010.26.949

On the distribution of periodic orbits

Katrin Gelfert 1, and  Christian Wolf 2,

1. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil
2. 

Department of Mathematics, Wichita State University, Wichita, Kansas, 67260

Received  January 2009 Revised  September 2009 Published  December 2009

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 Ptop$(\varphi)$ 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: equilibrium states, large deviation., topological pressure, non-uniformly hyperbolic dynamics, Pesin theory.
Mathematics Subject Classification: Primary: 37D40; Secondary: 37D50, 37C3.
Citation: Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949
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