Cesàro Convergence of Spherical Averages for Measure-Preserving Actions of Markov Semigroups and Groups

Cesàro convergence of spherical averages is proved for measure-preserving actions of Markov semigroups and groups. Convergence in the mean is established for functions in L^p, $1\leq p<\infty$, 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 Cesàro convergence of spherical averages with respect to any symmetric set of generators.