Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts

Abstract

Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$be a countable group of $\nu$-preserving invertible maps of $X$ intoitself. To a probability measure $\mu$ on $\Gamma$ corresponds arandom walk on $X$ with Markov operator $P$ given by $P\psi(x) =\sum_{a} \psi(ax) \, \mu(a)$. We consider various examples ofergodic $\Gamma$-actions and random walks and their extensions by avector space: groups of automorphisms or affine transformations oncompact nilmanifolds, random walks in random scenery on non amenablegroups, translations on homogeneous spaces of simple Lie groups,random walks on motion groups. A powerful tool in this study is thespectral gap property for the operator $P$ when it holds. We use itto obtain limit theorems, recurrence/transience property andergodicity for random walks on non compact extensions of thecorresponding dynamical systems.