Non-singular $\mathbb{Z}^d$-actions: an ergodic theorem over rectangles with application to the critical dimensions

TL;DR

This paper extends ergodic theorems to non-singular $bZ^d$-actions over rectangles with arbitrary growth, establishing invariants called critical dimensions that are preserved under metric isomorphism, with applications to product actions.

Contribution

It introduces a non-singular ergodic theorem for $bZ^d$-actions over rectangles with arbitrary side growth and demonstrates the invariance of critical dimensions under metric isomorphism.

Findings

Proved a non-singular ergodic theorem for $bZ^d$-actions over rectangles.

Established that critical dimensions are invariants of metric isomorphism.

Calculated critical dimensions for a class of product actions.

Abstract

We adapt techniques of Hochman to prove a non-singular ergodic theorem for $\mathbb{Z}^d$-actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for a class of product actions.