Chapter 2 Smooth ergodic theory and nonuniformly hyperbolic dynamics

Abstract

This chapter discusses smooth ergodic theory and nonuniformly hyperbolic dynamics. Smooth ergodic theory studies topological and ergodic properties of smooth dynamical systems with nonzero Lyapunov exponents. There are two classes of hyperbolic invariant measures on compact manifolds for which one can obtain a sufficiently complete description of its ergodic properties. They are: smooth measures, that is, measures that are equivalent to the Riemannian volume with the Radon-Nikodym derivative bounded from above and bounded away from zero, and Sinai-Ruelle-Bowen measures. Nonuniform hyperbolicity conditions can be expressed in terms of the Lyapunov exponents—that is, a dynamical system is nonuniformly hyperbolic if it admits an invariant measure with nonzero Lyapunov exponents almost everywhere. This provides an efficient tool in verifying the nonuniform hyperbolicity conditions and determines the importance of the nonuniform hyperbolicity theory in applications. The nonuniform hyperbolicity theory covers an enormous area of dynamics, such as nonuniformly hyperbolic one-dimensional transformations, random dynamical systems with nonzero Lyapunov exponents, billiards and related systems (for example, systems of hard balls), and numerical computation of Lyapunov exponents.