Mixing sequences for non-mixing transformations and group actions

TL;DR

This paper demonstrates the existence of non-mixing maps that exhibit mixing behavior along specific sequences, including those with the Rajchman dissociated property, and explores their implications for group actions and rigidity.

Contribution

It introduces new examples of non-mixing maps that are mixing along certain sequences, extending the understanding of mixing properties in dynamical systems and group actions.

Findings

Existence of non-mixing maps that are mixing along sequences with Rajchman dissociated property

Characterization of mixing sequences for weak mixing ${1}/{2}$-rigid transformations

Extension of results to infinite countable abelian group actions

Abstract

We establish that there are non-mixing maps that are mixing on appropriate sequences including sequences $(s_i)$ which satisfy the Rajchman dissociated property. Our examples are based on the staircase rank one construction, $M$-towers constructions and the Gaussian transformations. As a consequence, we obtain there are non-mixing maps which are mixing along the squares. We further prove that a sequence $M=(m_n)$ is a mixing sequence for some weak mixing ${1}/{2}$-rigid transformation $T$ if and only if the complement of $M$ is a thick set. This result is generalized to ${r}/{(r+1)}$-rigid transformations for $r\in \mathbb{N}$. Moreover, by applying Host-Parreau characterization of the set of continuity from Harmonic Analysis, we extend our results to the infinite countable abelian group actions.