On the work of Dolgopyat on partial and Nonuniform Hyperbolicity

This paper is a nontechnical survey and aims to illustrate Dolgo-pyat'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 hyper-bolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.