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

Bufetov, Aleksandr I.; Klimenko, Alexey V.; Khristoforov, Mikhail I.

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

Keywords:

Cesàro convergence; measure-preserving transformations; spherical averages; mean convergence; almost everywhere convergence; Markov semigroup