Cesaro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups

TL;DR

This paper proves Cesaro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups, including hyperbolic groups, in both mean and pointwise senses for various function spaces.

Contribution

It establishes new Cesaro convergence results for spherical averages in measure-preserving actions of Markov semigroups and hyperbolic groups, extending previous understanding.

Findings

Convergence in the mean for functions in L^p, 1 ≤ p < ∞.

Pointwise convergence for functions in L^∞.

Cesaro convergence for hyperbolic groups with symmetric generators.

Abstract

Cesaro convergence of spherical averages is proven for measure-preserving actions of Markov semigroups and groups. Convergence in the mean is established for functions in $L^p$, $1\le p<\infty$, and pointwise convergence for functions in $L^\infty$. In particular, for measure-preserving actions of word hyperbolic groups (in the sense of Gromov) we obtain Cesaro convergence of spherical averages with respect to any symmetric set of generators.