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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

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