Journal of Modern Dynamics (JMD)

On the work of Dolgopyat on partial and nonuniform hyperbolicity

Pages: 227 - 241, Issue 2, April 2010      doi:10.3934/jmd.2010.4.227

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Yakov Pesin - Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, United States (email)

Abstract: This paper is a nontechnical survey and aims to illustrate Dolgopyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyperbolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.

Keywords:  Brin prize, Dolgopyat, partial hyperbolicity, Lyapunov exponents, accessibility, stable ergodicity, SRB-measures.
Mathematics Subject Classification:  58F09.

Received: February 2010;      Revised: May 2010;      Available Online: August 2010.