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On Markov operators and ergodic theorems for group actions. (English) Zbl 1281.37004
From the introduction: “This paper is a brief survey of the method of Markov operators in the study of ergodic theorems for actions of free groups. This method is based on the following idea. Consider a measure-preserving action of a finitely-generated free group on a probability space. One can assign to this action a special stationary Markov process, whose asymptotic behaviour governs convergence properties of ergodic averages of the initial action.”
The simplest stationary Markov process that one can obtain from such an action is given by a random walk on the group, pushed down to the orbits on the probability space that supports the action. In the simplest cases of the method in the present paper, this must be modified by multiplying the state space by a finite list of symbols, and using these to give the random walk a record of which group element it followed at its previous step, so that it does not backtrack. After setting this up correctly, the authors explain how classical norm- and pointwise-convergence results for the Markov operator of this Markov process translate into the corresponding convergence of the spherical averages over the free group action, or of some natural variant of those spherical averages. Naturally defined ergodic averages for both free groups and free semigroups may be treated this way. This elegant method has been developed in several previous papers by Oseledets, Grigorchuk and the first author. It both recovers several known ergodic theorems in this setting (such as those of Nevo-Stein and Bowen-Nevo), and proves some new such results. The present paper is a clear and attractive exposition of this method; a few technical details are omitted, but it would serve as an excellent introduction to those other works.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
47A35 Ergodic theory of linear operators
28D05 Measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems
60J05 Discrete-time Markov processes on general state spaces
47B65 Positive linear operators and order-bounded operators
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