Multiple Mixing and Rank One Group Actions

del Junco, Andrés and Yassawi, Reem (2000). Multiple Mixing and Rank One Group Actions. Canadian Journal of Mathematics, 52(2) pp. 332–347.



Suppose G is a countable, Abelian group with an element of infinite order and let X be a mixing rank one action of G on a probability space. Suppose further that the Følner sequence {Fn} indexing the towers of X satisfies a “bounded intersection property”: there is a constant p such that each {Fn} can intersect no more than p disjoint translates of {Fn}. Then X is mixing of all orders. When G = 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 X is necessarily product measure. This method generalizes Ryzhikov’s technique.

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