American Institute of Mathematical Sciences

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October  2008, 20(4): 849-876. doi: 10.3934/dcds.2008.20.849

Physical measures at the boundary of hyperbolic maps

Vítor Araújo 1, and  Ali Tahzibi 2,

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, 21.945-970 Rio de Janeiro, RJ
2. 

Departamento de Matematica, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São Carlos-SP

Received  November 2006 Revised  October 2007 Published  January 2008

We consider diffeomorphisms of a compact manifold with a dominated splitting which is hyperbolic except for a "small" subset of points (Hausdorff dimension smaller than one, e.g. a denumerable subset) and prove the existence of physical measures and their stochastic stability. The physical measures are obtained as zero-noise limits which are shown to satisfy the Entropy Formula.
Keywords: physical measures, Dominated splitting, equilibrium states, partial hyperbolicity, stochastic stability., random perturbations.
Mathematics Subject Classification: Primary: 37D25. Secondary: 37D30, 37D2.
Citation: Vítor Araújo, Ali Tahzibi. Physical measures at the boundary of hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 849-876. doi: 10.3934/dcds.2008.20.849
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