Product recurrent properties, disjointness and weak disjointness

TL;DR

This paper investigates various forms of product recurrence and disjointness in dynamical systems, establishing new connections between recurrence properties, minimality, entropy, and disjointness with minimal systems.

Contribution

It introduces new results on the structure of weakly product recurrent points, their minimality, and disjointness conditions in dynamical systems.

Findings

Closure of syndetic-product recurrent points has dense minimal points

Piecewise syndetic-product recurrent points are minimal

Weakly mixing systems with dense minimal points are disjoint from all minimal PI systems

Abstract

Let $\F$ be a collection of subsets of $\Z_+$ and $(X,T)$ be a dynamical system. $x\in X$ is $\F$-recurrent if for each neighborhood $U$ of $x$, $\{n\in\Z_+:T^n x\in U\}\in \F$. $x$ is $\F$-product recurrent if $(x,y)$ is recurrent for any $\F$-recurrent point $y$ in any dynamical system $(Y,S)$. It is well known that $x$ is $\{infinite\}$-product recurrent if and only if it is minimal and distal. In this paper it is proved that the closure of a $\{syndetic\}$-product recurrent point (i.e. weakly product recurrent point) has a dense minimal points; and a $\{piecewise syndetic\}$-product recurrent point is minimal. Results on product recurrence when the closure of an $\F$-recurrent point has zero entropy are obtained. It is shown that if a transitive system is disjoint from all minimal systems, then each transitive point is weakly product recurrent. Moreover, it proved that each weakly mixing system with dense minimal points is disjoint from all minimal PI systems; and each weakly mixing system with a dense set of distal points or an $\F_s$-independent system is disjoint from all minimal systems. Results on weak disjointness are described when considering disjointness.