A backward ergodic theorem along trees and its consequences for free group actions

TL;DR

This paper establishes a new backward ergodic theorem for free group actions, extending classical results by averaging over trees in the Cayley graph, with applications to boundary actions and Markov measures.

Contribution

It introduces a novel backward ergodic theorem for countable-to-one transformations, strengthening previous free group ergodic theorems and confirming Bufetov's 2002 conjecture.

Findings

Proves a pointwise ergodic theorem for free group actions over finite subtrees.

Develops a backward ergodic theorem for transformations with tree-like pasts.

Applies the theorem to boundary actions and Markov measures, demonstrating broad utility.

Abstract

We prove a new pointwise ergodic theorem for probability-measure-preserving (pmp) actions of free groups, where the ergodic averages are taken over arbitrary finite subtrees of the standard Cayley graph rooted at the identity. This result is a significant strengthening of a theorem of Grigorchuk (1987) and Nevo and Stein (1994), and a version of it was conjectured by Bufetov in 2002. Our theorem for free groups arises from a new - backward - ergodic theorem for a countable-to-one pmp transformation, where the averages are taken over arbitrary trees of finite height in the backward orbit of the point (i.e. trees of possible pasts). We also discuss other applications of this backward theorem, in particular to the shift map with Markov measures, which yields a pointwise ergodic theorem along trees for the boundary actions of free groups.