Hyperbolic invariant sets with positive measures

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August 2006, 15(3): 811-818. doi: 10.3934/dcds.2006.15.811

Hyperbolic invariant sets with positive measures

1.	Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Received July 2005 Revised January 2006 Published April 2006

Abstract
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In this short paper we prove some results concerning volume-preserving Anosov diffeomorphisms on compact manifolds. The first theorem is that if a $C^{1 + \alpha}$, $\alpha >0$, volume-preserving diffeomorphism on a compact connected manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. The same result had been obtained by Bochi and Viana [2]. This result is not necessarily true for $C^1$ maps. The proof uses a Pugh-Shub type of dynamically defined measure density points, which are different from the standard Lebesgue density points. We then give a direct proof of the ergodicity of $C^{1+\alpha}$ volume preserving Anosov diffeomorphisms, without using the usual Hopf arguments or the Birkhoff ergodic theorem. The method we introduced also has interesting applications to partially hyperbolic and non-uniformly hyperbolic systems.

Keywords: Ergodic theory, hyperbolic invariant set., volume-preserving diffeomorphisms, Anosov map.

Mathematics Subject Classification: 37A25, 37D05, 37D2.

Citation: Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811

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