Mixing sequences for non-mixing transformations and 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.

arXiv comment23 pages. This is our contribution to the question raised by Professor Vitaly Bergelson. The paper is dedicated to the 80th Anniversary of Professor Jean-Paul Thouvenot. Scientific suggestions and comments with questions are welcome