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Multiple Mixing and Rank One Group Actions

  Published:2000-04-01
 Printed: Apr 2000
  • Andrés del Junco
  • Reem Yassawi
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Abstract

Suppose $G$ is a countable, Abelian group with an element of infinite order and let ${\cal X}$ be a mixing rank one action of $G$ on a probability space. Suppose further that the F\o lner sequence $\{F_n\}$ indexing the towers of ${\cal X}$ 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 ${\cal X}$ is mixing of all orders. When $G={\bf Z}$, 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 ${\cal X}$ is necessarily product measure. This method generalizes Ryzhikov's technique.
MSC Classifications: 28D15 show english descriptions General groups of measure-preserving transformations 28D15 - General groups of measure-preserving transformations
 

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