A horospherical ratio ergodic theorem for actions of free groups

TL;DR

This paper establishes a ratio ergodic theorem for free group actions, extending ergodic theory to non-abelian free groups and horospherical averages, with implications for non-singular dynamics.

Contribution

It introduces a ratio ergodic theorem for free group actions and horospheres, advancing the understanding of non-abelian group dynamics in ergodic theory.

Findings

Proves a ratio ergodic theorem for amenable equivalence relations.

Establishes a ratio ergodic theorem for free group actions along horospheres.

Extends ergodic theorems to non-abelian free groups.

Abstract

We prove a ratio ergodic theorem for amenable equivalence relations satisfying a strong form of the Besicovich covering property. We then use this result to study general non-singular actions of non-abelian free groups and establish a ratio ergodic theorem for averages along horospheres.