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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