## Cesàro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups.(English)Zbl 1267.22003

The authors prove, in their main theorem (Theorem 1), that the sequence of Cesaro averages of the spherical averages, of a Markov semigroup $$\Gamma$$ with respect to a finite generating set, converges. The result in Theorem 1 provides the mean convergence of the sequence of Cesaro averages for functions in $$L^p$$ and pointwise convergence for functions in $$L^{\infty}$$. The Markov semigroup $$\Gamma$$ is assumed to act by measure-preserving transformations on a probability space. Therefore, the result in Theorem 1 will apply to all measure-preserving actions of word hyperbolic groups.

### MSC:

 22D40 Ergodic theory on groups 28D05 Measure-preserving transformations

### Keywords:

Marko semigroup; Cesaro convergence; spherical averages
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