Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix \(\varPi \) defining our Markov chain. Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains.
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The authors are deeply grateful to Vadim Kaimanovich for useful discussions. Lewis Bowen is supported in part by NSF grant DMS-0968762, NSF CAREER Award DMS-0954606 and BSF grant 2008274. Alexander Bufetov’s research is carried out thanks to the support of the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the programme “Investissements d’Avenir ” of the Government of the French Republic, managed by the French National Research Agency (ANR). Bufetov is also supported in part by the Grant MD-2859.2014.1 of the President of the Russian Federation, by the Programme “Dynamical systems and mathematical control theory” of the Presidium of the Russian Academy of Sciences, and by the RFBR grant 13-01-12449. Olga Romaskevich is supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). She is as well supported by RFBR project 13-01-00969-a and Poncelet Laboratory.
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Bowen, L., Bufetov, A. & Romaskevich, O. Mean convergence of Markovian spherical averages for measure-preserving actions of the free group. Geom Dedicata 181, 293–306 (2016). https://doi.org/10.1007/s10711-015-0124-2
- Ergodic theorems
- Markov operators
- Spherical averages
- Free groups
Mathematics Subject Classification (2010)