Smooth ergodic theory of random dynamical systems
Smooth ergodic theory investigates the long-time behaviour of the iterations \(f^n\), \(n \in \mathbb{N}\) (or \(n \in \mathbb{Z})\), of a smooth map \(f : M \to M\), where \(M\) is a manifold, giving relations between invariant measures and the Lebesgue measure class of \(M\), invariant manifolds, entropy, Lyapunov exponents and others. The present volume investigates not iterates of a single smooth map \(f\), but replaces \(f\) by a probability law \(\nu\) on the space of smooth maps. Then products of maps \(f_n, \dots, f_1\) are considered, where \((f_i)_i\) are independent random variables distributed according to \(\nu\). This is a special case of a random dynamical system. The book can be considered as a followup to \textit{Y. Kifer}'s book [Ergodic theory of random transformations, Birkhäuser, Boston (1986; Zbl 0604.28014)]. The present work is centered around the following result: The entropy associated with an arbitrary invariant measure for the system is bounded from above by the sum of the positive Lyapunov exponents associated with this invariant measure. Furthermore, equality holds if and only if the invariant measure is `absolutely continuous along unstable manifolds' (also addressed to as the Sinai-Bowen-Ruelle or SBR property). The table of contents reads as follows. Chapter 0: Preliminaries. Chapter I: Entropy and Lyapunov Exponents of Random Diffeomorphisms. Chapter II: Estimation from Above Through Lyapunov Exponents. Chapter III: Stable Invariant Manifolds of Random Diffeomorphisms. Chapter IV: Estimation of Entropy from Below Through Lyapunov Exponents. Chapter V: Stochastic Flows of Diffeomorphisms. Chapter VI: Characterization of Measures Satisfying Entropy Formula. Chapter VII: Random Perturbations of Hyperbolic Attractors. Appendix: A Margulis-Ruelle Inequality for Random Dynamical Systems.
scientific article; zbMATH DE number 803191