Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. (Statistical equilibrium for products of independent random diffeomorphisms)
type: Item
instance of: scholarly article
title: Équilibre statistique pour les produits de difféomorphismes aléatoires indépendants. (Statistical equilibrium for products of independent random diffeomorphisms)
zbMATH Open document ID: 0614.60047
author: Yves Le Jan
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77288
review text: Let V be a compact Riemannian variety, G be the group of diffeomorphisms on V, \(\nu\) be a probability on G, and finally \(\Pi\) (V) be the set of probabilities on V. Define a Markov kernel P, assumed ergodic, on V by \(Pf(x)=\int f(Tx)\nu (dT)\), and let m denote a P-invariant element of \(\Pi\) (V). On \(\Omega =G^ Z\) define the coordinate mappings \(T_ i\), and let \(P=\nu^ Z.\) The author investigates structural properties of the random measure \(\mu\) on V defined by \[ \mu (\omega)=\lim_{n}m[T_ 0\circ...\circ T_{- n}(\omega)]^{-1}. \] For example, let \(\lambda\) and \(\alpha\) be, respectively, the largest Lyapunov exponent and the sum of the exponents of the system; then if \(\alpha <0\), \(\mu\) is singular with respect to \(\rho\), the Riemannian volume on V, P-a.s., while if \(m\sim \rho\), then \(\alpha\leq 0\). Similarly, if \(\lambda <0\), then almost surely \(\mu\) is a finite sum of point measures of the same mass, i.e., \(\mu =(1/K)\sum^{K}_{i=1}\epsilon_{X_ i}\), where K is a finite random variable, the \(X_ i\) are random elements of V, and \(\epsilon_ x\) is the point mass at x.
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60B05
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 3994665
zbMATH Keywords: group of diffeomorphisms
zbMATH Keywords: structural properties of the random measure
zbMATH Keywords: largest Lyapunov exponent
reviewed by: Alan F. Karr
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3994665