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

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