Rigidity, weak mixing, and recurrence in abelian groups

Ethan M. Ackelsberg

doi:10.3934/dcds.2021168

Article Contents

2022, Volume 42, Issue 4: 1669-1705. Doi: 10.3934/dcds.2021168

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Rigidity, weak mixing, and recurrence in abelian groups

Ethan M. Ackelsberg

Department of Mathematics, Ohio State University, Columbus, OH 43210, USA

Received: July 2020

Revised: September 2021

Early access: November 2021

Published: April 2022

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Abstract

The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about $ \mathbb{Z} $-actions extend to this setting:

1. If $ (a_n) $ is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.

2. There exists a sequence $ (r_n) $ such that every translate is both a rigidity sequence and a set of recurrence.

The first of these results was shown for $ \mathbb{Z} $-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in $ \mathbb{Z} $ by Griesmer [23]. While techniques for handling $ \mathbb{Z} $-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.

As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in $ \mathbb{Z} $, while others exhibit new phenomena.

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Mathematics Subject Classification: Primary: 37A15; Secondary: 37A25, 37B20, 37A45.

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