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Multiple Mixing and Rank One Group Actions

Published online by Cambridge University Press:  20 November 2018

Andrés del Junco  and
Reem Yassawi
Andrés del Junco
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, M5S 1A1
Reem Yassawi
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6
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Abstract

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Suppose $G$ is a countable, Abelian group with an element of infinite order and let $\text{ }\!\!\chi\!\!\text{ }$ be a mixing rank one action of $G$ on a probability space. Suppose further that the Følner sequence $\{{{F}_{n}}\}$ indexing the towers of $\text{ }\!\!\chi\!\!\text{ }$ satisfies a “bounded intersection property”: there is a constant $p$ such that each $\{{{F}_{n}}\}$ can intersect no more than $p$ disjoint translates of $\{{{F}_{n}}\}$. Then $\text{ }\!\!\chi\!\!\text{ }$ is mixing of all orders. When $G\,=\,\mathbf{Z}$, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of $\text{ }\!\!\chi\!\!\text{ }$ is necessarily product measure. This method generalizes Ryzhikov’s technique.

Keywords

28D15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 2 , 01 April 2000 , pp. 332 - 347
Copyright
Copyright © Canadian Mathematical Society 2000

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