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Ergodic theorems for actions of hyperbolic groups. (English) Zbl 1266.28008

The authors give a short proof of the almost everywhere convergence for the Cesàro averages of the spherical averages for \({L_\infty}\) functions with respect to measure-preserving actions of word hyperbolic groups. This result recently is also obtained by A. I. Bufetov, A. V. Klimenko and M. I. Khristoforov [Russ. Math. Surv. 66, No. 3, 633–634 (2011); translation from Usp. Mat. Nauk. 66, No. 3, 203–204 (2011; Zbl 1244.37006)].
The proof is based on an association to strongly Markov groups (word hyperbolic groups are the particular case of such groups) of some directed graph. This allows to reduce the problem to the ergodic theorem for the Markov matrix that encodes a word hyperbolic group. The structure of such a matrix observed by D. Calegari and K. Fujiwara [Ergodic Theory Dyn. Syst. 30, No. 5, 1343–1369 (2010; Zbl 1217.37025)] is used in the proof.
At the end of the paper, an application to linear actions of discrete groups on the plane is considered.

MSC:

28D15 General groups of measure-preserving transformations
37A15 General groups of measure-preserving transformations and dynamical systems
37A30 Ergodic theorems, spectral theory, Markov operators
60J05 Discrete-time Markov processes on general state spaces
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