Cesaro convergence of spherical averages for measure-preserving actions   of Markov semigroups and groups
  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. 
Alexander Bufetov, Mikhail Khristoforov, Alexey Klimenko
math.DS math.GR math.OA