Uniformly ergodic probability measures

TL;DR

This paper characterizes when probability measures on locally compact groups exhibit uniform ergodicity and mixing, linking these properties to the group's compactness, measure support, and spectral characteristics.

Contribution

It provides a complete characterization of uniform ergodicity and mixing for convolution operators on locally compact groups, establishing conditions on the measure and the group structure.

Findings

Uniform ergodicity occurs iff the group is compact, the measure is adapted, and 1 is isolated in the spectrum.

Uniform complete mixing occurs iff the group is compact, the measure is spread-out, and only 1 is a unimodular spectral value.

No difference exists between the operators and their restrictions regarding uniform ergodicity.

Abstract

Let $G$ be a locally compact group and $\mu$ be a probability measure on $G$. We consider the convolution operator $\lambda_1(\mu)\colon L_1(G)\to L_1(G)$ given by $\lambda_1(\mu)f=\mu \ast f$ and its restriction $\lambda_1^0(\mu)$ to the augmentation ideal $L_1^0(G)$. Say that $\mu$ is uniformly ergodic if the Ces\`aro means of the operator $\lambda_1^0(\mu)$ converge uniformly to 0, that is, if $\lambda_1^0(\mu)$ is a uniformly mean ergodic operator with limit 0 and that $\mu$ is uniformly completely mixing if the powers of the operator $\lambda_1^0(\mu)$ converge uniformly to 0. We completely characterize the uniform mean ergodicity of the operator $\lambda_1(\mu)$ and the uniform convergence of its powers and see that there is no difference between $\lambda_1(\mu)$ and $\lambda_1^0(\mu) $ in this regard. We prove in particular that $\mu$ is uniformly ergodic if and only if $G$ is compact, $\mu$ is adapted (its support is not contained in a proper closed subgroup of $G$) and 1 is an isolated point of the spectrum of $\mu$. The last of these three conditions is actually equivalent to $\mu$ being spread-out (some convolution power of $\mu$ is not singular). The measure $\mu$ is uniformly completely mixing if and only if $G$ is compact, $\mu$ is spread-out and the only unimodular value of the spectrum of $\mu$ is 1.