# American Institute of Mathematical Sciences

June  2006, 16(2): 505-512. doi: 10.3934/dcds.2006.16.505

## On hyperbolic measures and periodic orbits

 1 Department of Mathematics, Rice University, Houston, TX 77005, United States

Received  September 2005 Revised  September 2005 Published  July 2006

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.
Citation: Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
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