Geodesic
Maximal inequality and ergodic theorems for Markov groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 33-48

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The paper shows that for actions of Markov semigroups, in particular, of finitely generated word hyperbolic groups, the Cesàro means of spherical averages converge almost everywhere for any function from the class $L^p$, $p>1$. 
@article{TM_2012_277_a2,
     author = {A. I. Bufetov and A. V. Klimenko},
     title = {Maximal inequality and ergodic theorems for {Markov} groups},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {33--48},
     publisher = {mathdoc},
     volume = {277},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2012_277_a2/}
}
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A. I. Bufetov; A. V. Klimenko. Maximal inequality and ergodic theorems for Markov groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical control theory and differential equations, Tome 277 (2012), pp. 33-48. http://geodesic.mathdoc.fr/item/TM_2012_277_a2/