A violation of multiple mixing close to an extremal

Tikhonov, S.

doi:10.1090/mosc/322

A violation of multiple mixing close to an extremal
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by S. V. Tikhonov
Translated by: A. I. Shtern

Trans. Moscow Math. Soc. 2021, 173-181

DOI: https://doi.org/10.1090/mosc/322

Published electronically: March 15, 2022

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Abstract:

Given a mixing action $L$ of a group $G$ and a set $A$ of half measure we consider the possible limits of the measures $\mu (A\cap L^{m_{i}}A\cap L^{n_{i}}A)$ as $i\to \infty$ and $m_{i},n_{i},m_{i}-n_{i}\to \infty$. If the action is 3-mixing, then these limits are always equal to $1/8$. In the Ledrappier example, this limit is zero for some sequences. The following question is studied: what can be said about actions if one of these limits is positive but small? In the paper we make several observations on this topic.

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