Sharp ergodic theorems for group actions and strong ergodicity

Let $\mu$ be a probability measure on a locally compact
group $G$, and suppose $G$ acts 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 thorems
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 bring many natural examples, among them automorphism actions
on tori and actions on Lie groups' homogeneous spaces, for which our results
can be applied.

Bibliographical: Ergodic Theory Dynam. Systems

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[Papers] [Alex Furman]