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

Anthony H. Dooley; Kieran Jarrett

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arXiv · 1606.01620

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

Anthony H. Dooley·Kieran Jarrett

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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.

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BibTeX RIS

Anthony H. Dooley, Kieran Jarrett. 2016-06-06. Non-singular $\mathbb{Z}^d$-actions: an ergodic theorem over rectangles with application to the critical dimensions. https://arxiv.org/abs/1606.01620

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