On weak product recurrence and synchronization of return times

TL;DR

This paper investigates product recurrence in dynamical systems, establishing equivalences between different recurrence notions, analyzing return times of weakly mixing sets, and providing new insights into disjointness properties.

Contribution

It proves the equivalence of certain recurrence notions with product recurrence, extends characterization of return times of distal points, and offers new conditions for weak product recurrence.

Findings

$_{ps}-PR$ and $_{pubd}-PR$ are equivalent to product recurrence

Characterization of return times of distal points is extended

Weakly mixing systems are disjoint from all minimal distal systems

Abstract

The paper is devoted to study of product recurrence. First, we prove that notions of $\F_{ps}-PR$ and $\F_{pubd}-PR$ are exactly the same as product recurrence, completing that way results of [P. Dong, S. Shao and X. Ye, \emph{Product recurrent properties, disjointness and weak disjointness}, Israel J. Math.], and consequently, extending the characterization of return times of distal points which originated from works of Furstenberg. We also study the structure of the set of return times of weakly mixing sets. As a consequence, we obtain new sufficient conditions for $\F_{s}-PR$ and also find a short proof that weakly mixing systems are disjoint with all minimal distal systems (in particular, our proof does not involve Furstenberg's structure theorem of minimal distal systems).