Geometric and ergodic theory of hyperbolic dynamical systems

The topic of this survey lies at the confluence of two branches of dynamical systems, the geometric theory, of ordinary differential epuations and ergodic theory. The first goes back to Poincare, who pioneered the use of geometric methods in dynamical systems in his work on celestial mechanics; the latter goes back to Boltzmann, whose ideas are part of the foundation of modern statistical mechanics. Hyperobolicity in dynamical systems is a geometric condition describing the exponential divergence of nearby orbits. It leads to irregular, chaotic and unpredictable patterns of behavior. Along with quasi-periodic dynamics or KAM theory, which lies at the opposite end of the ordered-disordered spectrum of dynamical behaviors, hyperbolic theory is one of the better understood areas of dynamical systems today. This article is about the geometric and ergodic theory of hyperbolic systems. My aim here is not to give a complete list of all the important results, but to focus on a few areas of activity and to describe in as coherent a fashion as I can some of the progress over the last 20-30 years. The topics I have chosen are I. Billiards and related physical systems II. Analysis of a class of strange attractors III. Entropy, Lyapunov exponents and dimension IV. Correlation decay and related statistical properties. This article is intended for the broader mathematics community as a ell as researchers in dynamical systems. Same background material is included at the beginning for readers not in dynamics.