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

Part of: Ergodic theory

Published online by Cambridge University Press:  02 December 2020

ANTHONY H. DOOLEY  and
KIERAN JARRETT
ANTHONY H. DOOLEY
Affiliation:
School of Mathematical and Physical Sciences, University of Technology Sydney, NSW2007, Australia (e-mail: Anthony.Dooley@uts.edu.au)
KIERAN JARRETT*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK
*
e-mail: K.Jarrett@bath.ac.uk
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Abstract

We adapt techniques developed by 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 the natural action of $\mathbb {Z}^d$ on a product of d measure spaces.

Keywords

classical ergodic theorygroup actionscritical dimensions

MSC classification

Primary: 37A40: Nonsingular (and infinite-measure preserving) transformations
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 37A15: General groups of measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3722 - 3739
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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