Convergence of spherical averages for actions of Fuchsian Groups

TL;DR

This paper proves pointwise convergence of spherical averages for Fuchsian group actions using a novel symbolic coding that encodes shortest paths and creates a reversible Markov chain, extending previous methods to a broader class of groups.

Contribution

It introduces a new symbolic coding for Fuchsian groups that enables convergence proofs for spherical averages, generalizing techniques from free groups.

Findings

Proves pointwise convergence of spherical averages for Fuchsian groups.

Develops a reversible Markov chain model for the group action.

Extends convergence results from free groups to Fuchsian groups.

Abstract

Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.