On hyperbolic measures and periodic orbits

,Department Of Mathematics, Rice University, Houston, Tx 77005 ,Ilie Ugarcovici

doi:10.3934/dcds.2006.16.505

,Department Of Mathematics, Rice University, Houston, Tx 77005 , Ilie Ugarcovici

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https://doi.org/10.3934/dcds.2006.16.505

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Journal: Discrete & Continuous Dynamical Systems - A	Publication Date: Jan 1, 2006
Citations: 11	License type: cc-by

Abstract

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.

Highlights

The theory of nonuniformly hyperbolic dynamical systems, often called Pesin theory due to the fundamental work of Ya
In [6], Katok proved that if the measure is nonatomic, its support is contained in the closure of the hyperbolic periodic points that have transverse homoclinic points
The support of the measure is contained in the closure of all such hyperbolic periodic points

Summary

Introduction

The theory of nonuniformly hyperbolic dynamical systems, often called Pesin theory due to the fundamental work of Ya. Pesin in the mid-seventies ([16, 17]), studies smooth dynamical systems preserving a hyperbolic measure, i.e., a measure whose Lyapunov exponents are nonzero almost everywhere In his seminal work [6], Anatole Katok established several essential results about the periodic orbits of nonuniformly hyperbolic dynamical systems. He proved the closing property for such systems, found that the asymptotic exponential growth of periodic points of a diffeomorphism preserving a hyperbolic probability measure is bounded from below by its measure-theoretic entropy, and proved the existence of periodic orbits with transversal homoclinic intersections ( the existence of uniform hyperbolic horseshoes).

ILIE UGARCOVICI

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