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

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.

26 pages, submitted to Ergodic Theory and Dynamical Systems

References in corpus (1)

• A ratio ergodic theorem for multiparameter non-singular actions