Transitive points via Furstenberg family

Abstract

Let $(X,T)$ be a topological dynamical system and $\mathcal{F}$ be a Furstenberg family (a collection of subsets of $\mathbb{Z}_+$ with hereditary upward property). A point $x\in X$ is called an $\mathcal{F}$-transitive one if $\{n\in\mathbb{Z}_+:\, T^n x\in U\}\in\F$ for every nonempty open subset $U$ of $X$; the system $(X,T)$ is called $\F$-point transitive if there exists some $\mathcal{F}$-transitive point. In this paper, we aim to classify transitive systems by $\mathcal{F}$-point transitivity. Among other things, it is shown that $(X,T)$ is a weakly mixing E-system (resp.\@ weakly mixing M-system, HY-system) if and only if it is $\{\textrm{D-sets}\}$-point transitive (resp.\@ $\{\textrm{central sets}\}$-point transitive, $\{\textrm{weakly thick sets}\}$-point transitive). It is shown that every weakly mixing system is $\mathcal{F}_{ip}$-point transitive, while we construct an $\mathcal{F}_{ip}$-point transitive system which is not weakly mixing. As applications, we show that every transitive system with dense small periodic sets is disjoint from every totally minimal system and a system is $\Delta^*(\mathcal{F}_{wt})$-transitive if and only if it is weakly disjoint from every P-system.