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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
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