Smooth Ergodic Theory of Random Dynamical Systems

This book studies ergodic-theoretic aspects of random dynam- ical systems, i.e. of deterministic systems with noise. It aims to present a systematic treatment of a series of recent results concerning invariant measures, entropy and Lyapunov exponents of such systems, and can be viewed as an update of Kifer's book. An entropy formula of Pesin's type occupies the central part. The introduction of relation numbers (ch.2) is original and most methods involved in the book are canonical in dynamical systems or measure theory. The book is intended for people interested in noise-perturbed dynam- ical systems, and can pave the way to further study of the subject. Reasonable knowledge of differential geometry, measure theory, ergodic theory, dynamical systems and preferably random processes is assumed.

Preliminaries.- Entropy and Lyapunov exponents of random diffeomorphisms.- Estimation of entropy from above through Lyapunov exponents.- Stable invariant manifolds of random diffeomorphisms.- Estimation of entropy from below through Lyapunov exponents.- Stochastic flows of diffeomorphisms.- Characterization of measures satisfying entropy formula.- Random perturbations of hyperbolic attractors.