## Cesàro convergence of spherical averages for Markov groups and semigroups.(English. Russian original)Zbl 1244.37006

Russ. Math. Surv. 66, No. 3, 633-634 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 203-204 (2011).
Summary: Let $$\Gamma$$ be a finitely generated group. A choice of a finite generating set $$O$$ endows $$\Gamma$$ with a norm such that the norm $$|g|_O$$ of an element $$g\in\Gamma$$ is the length of the shortest word in the alphabet $$O$$ that represents $$g$$. We let $$S_O(n)=\{g:|g|_O= n\}$$.
Suppose that $$\Gamma$$ acts by measure-preserving transformations $$T_g$$, $$g\in\Gamma$$, on a probability space $$(X,\nu)$$. For any function $$\varphi\in L^1(X,\nu)$$ we consider the sequence of its spherical averages $s_n(\varphi)=\frac{1}{\#S_O(n)}\sum_{g\in S_O (n)}\varphi\circ T_g$ (here $$\#$$ denotes the cardinality of a finite set); if $$S_O(n)=\emptyset$$, we put $$s_n(\varphi)=0$$. Further, let us consider the Cesàro means of the spherical averages: $c_N(\varphi)= \frac 1N\sum^{N-1}_{n=0}s_n(\varphi).$ The main result of the paper establishes mean convergence of the sequence $$c_N(\varphi)$$ for any $$\varphi\in L^1(X,\nu)$$ and almost everywhere convergence of this sequence for $$\varphi\in L^\infty(X,\nu)$$ in the case when $$\Gamma$$ is a Markov semigroup with respect to the generating set $$O$$.

### MSC:

 37A30 Ergodic theorems, spectral theory, Markov operators 28D15 General groups of measure-preserving transformations 37A05 Dynamical aspects of measure-preserving transformations
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