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Ergodic Theory and Dynamical Systems Ergodic Theory and Dynamical Systems

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Sharp ergodic theorems for group actions and strong ergodicity

Published online by Cambridge University Press:  02 April 2001

ALEX FURMAN  and
YEHUDA SHALOM
ALEX FURMAN
Affiliation:
The Institute of Mathematics, The Hebrew University, Givat-Ram, Jerusalem 91904, Israel Current address: Mathematics department (m/c 249), University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA (e-mail: furman@math.uic.edu)
YEHUDA SHALOM
Affiliation:
The Institute of Mathematics, The Hebrew University, Givat-Ram, Jerusalem 91904, Israel Current address: Mathematics Department, Yale University, 10 Hillhouse Avenue, P.O. Box 208283, New Haven, CT 06520-8283, USA (e-mail: yehuda@math.yale.edu)
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Abstract

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.

The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 19 , Issue 4 , August 1999 , pp. 1037 - 1061
Copyright
1999 Cambridge University Press

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