Smooth ergodic theory of random dynamical systems
type: Item
instance of: scholarly article
title: Smooth ergodic theory of random dynamical systems
zbMATH Open document ID: 0841.58041
author: Min Qian
author: Peidong Liu
published in: Lecture Notes in Mathematics
publication date: 1995-10-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 803191
zbMATH Keywords: invariant measure
zbMATH Keywords: entropy
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: SBR-measure
zbMATH Keywords: Margulis-Ruelle inequality
reviewed by: Hans Crauel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0094308
OpenAlex ID: W647919729
DOI: 10.1007/BFB0094308
description: scientific article; zbMATH DE number 803191