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May  2010, 27(2): 589-607. doi: 10.3934/dcds.2010.27.589

Open problems in the theory of non-uniform hyperbolicity

Yakov Pesin 1, and  Vaughn Climenhaga 1,

1. 

Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, United States

Received  October 2009 Revised  February 2010 Published  February 2010

This is a survey-type article whose goal is to review some recent developments in studying the genericity problem for non-uniformly hyperbolic dynamical systems with discrete time on compact smooth manifolds. We discuss both cases of systems which are conservative (preserve the Riemannian volume) and dissipative (possess hyperbolic attractors). We also consider the problem of coexistence of hyperbolic and regular behaviour.
Keywords: SRB measures, $u$-measures., Non-uniform hyperbolicity, generic properties, mixed hyperbolicity, dissipative systems, conservative systems.
Mathematics Subject Classification: Primary: 37-02, 37D25, 37C20; Secondary: 37A10, 37D3.
Citation: Yakov Pesin, Vaughn Climenhaga. Open problems in the theory of non-uniform hyperbolicity. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 589-607. doi: 10.3934/dcds.2010.27.589
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