On the work of Dolgopyat on partial and nonuniform hyperbolicity

Yakov Pesin

doi:10.3934/jmd.2010.4.227

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2010, Volume 4, Issue 2: 227-241. Doi: 10.3934/jmd.2010.4.227 open access

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On the work of Dolgopyat on partial and nonuniform hyperbolicity (Brin Prize article)

Yakov Pesin^1,

1.
Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802

Received: January 31, 2010

Revised: April 30, 2010

Published: July 2010

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Abstract

This paper is a nontechnical survey and aims to illustrate Dolgopyat'sprofound contributions to smooth ergodic theory. I will discuss some ofDolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity withemphasis on the interaction between the two-the class of dynamical systemswith mixed hyperbolicity. On one hand, this includes uniformlypartially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in thecenter direction. The study of their ergodic properties has provided analternative approach to the Pugh-Shub stable ergodicity theory for bothconservative and dissipative systems. On the other hand, ideas of mixedhyperbolicity have been used in constructing volume-preservingdiffeomorphisms with nonzero Lyapunov exponents on any manifold.

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Mathematics Subject Classification: 58F09.

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