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.