Mean convergence of Markovian spherical averages for measure-preserving actions of the free group

TL;DR

This paper proves the mean convergence of Markovian spherical averages for measure-preserving actions of free groups, extending previous results to a broader class of Markov chains with less restrictive conditions.

Contribution

It introduces generalized Markov chains on generators and establishes mean convergence under mild nondegeneracy assumptions, broadening the scope beyond symmetric chains.

Findings

Mean convergence of spherical averages for a wider class of Markov chains.

Triviality of the tail sigma-algebra of the Markov operator.

Convergence holds under inequalities rather than equalities.

Abstract

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 $\Pi$ 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.