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Degenerate random perturbations of Anosov diffeomorphisms

Published online by Cambridge University Press:  17 July 2009

TATIANA YARMOLA
TATIANA YARMOLA*
Affiliation:
Department of Mathematics, Mathematics Building, University of Maryland, College Park, MD 20742-4015, USA (email: [email protected])
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Abstract

This paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank-k random perturbations of C2 diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms, hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimensional disks that support the distributions to certain foliations that arise from Anosov diffeomorphisms. We conclude that generic rank-k random perturbations have absolutely continuous invariant measures.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 3 , June 2010 , pp. 931 - 951
Copyright
Copyright © Cambridge University Press 2009

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