A ratio ergodic theorem for multiparameter non-singular actions

TL;DR

This paper establishes a ratio ergodic theorem for non-singular actions of multi-dimensional groups, utilizing geometric measure theory and covering lemmas to handle boundary concentration issues.

Contribution

It introduces a new ratio ergodic theorem for multiparameter non-singular actions along arbitrary norm balls, linking geometric properties to ergodic behavior.

Findings

Proves a ratio ergodic theorem for $Z^d$ and $R^d$ actions.

Connects the Besicovitch covering property to maximal inequalities.

Shows boundaries of balls have lower dimension, aiding the proof.

Abstract

We prove a ratio ergodic theorem for non-singular free $Z^d$ and $R^d$ actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in $R^d$. The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group actions, the Besicovitch covering property not only implies the maximal inequality, but is equivalent to it, implying that further generalization may require new methods.