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 $\{\text{D-sets}\}$-point transitive (resp.\@ $\{\text{central sets}\}$-point transitive, $\{\text{weakly thick sets}\}$-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 ${\Delta }^{\ast }({\mathcal{F}}_{wt})$-transitive if and only if it is weakly disjoint from every P-system.

Minor changes, 19 pages,to appear in Topology and its applications

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• Localization of mixing property via Furstenberg families
• Recurrence properties and disjointness on the induced spaces
• Dynamical compactness and sensitivity
• Types of mixings and transitivities in topological dynamics