Geodesic
An Ergodic Theorem for the Action of a~Free Semigroup
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 119-133

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An individual ergodic theorem for the action of a free semigroup is proved under the assumption that the measure is stationary. The proof involves the constructions of the associated stationary Markov process and of the skew shift. 
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     author = {R. I. Grigorchuk},
     title = {An {Ergodic} {Theorem} for the {Action} of {a~Free} {Semigroup}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {119--133},
     publisher = {mathdoc},
     volume = {231},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2000_231_a4/}
}
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R. I. Grigorchuk. An Ergodic Theorem for the Action of a~Free Semigroup. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 119-133. http://geodesic.mathdoc.fr/item/TM_2000_231_a4/