Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-07-03T09:27:20.155Z Has data issue: false hasContentIssue false
  • English
  • Français

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: yarmola@math.umd.edu)
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Choquet, G.. Lectures on Analysis. Vol. 1. W. A. Benjamin, New York, 1969.Google Scholar
[2]Franks, J. M.. Invariant sets of hyperbolic toral automorphisms. Amer. J. Math. 99(5) (1977), 1089–1095.CrossRefGoogle Scholar
[3]Hancock, S. G.. Construction of invariant sets for anosov diffeomorphisms. J. London Math. Soc. (2) 18(2) (1978), 339–348.CrossRefGoogle Scholar
[4]Kifer, Yu.. Random Perturbations of Dynamical Systems (Progress in Probability and Statistics, 16). Birkhäuser, Boston, MA, 1988.CrossRefGoogle Scholar
[5]Nualart, D.. The Malliavin Calculus and Related Topics (Probability and its Applications). Springer, Berlin, 2006.Google Scholar
[6]Robinson, C.. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (Studies in Advanced Mathematics). CRC Press, Boca Raton, FL, 1995.Google Scholar