Product recurrent properties, disjointness and weak disjointness

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.

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