Mathematics – Dynamical Systems
Scientific paper
2011-03-17
Topology and its Applications 158 (2011) pp. 2221-2231
Mathematics
Dynamical Systems
Minor changes, 19 pages,to appear in Topology and its applications
Scientific paper
10.1016/j.topol.2011.07.013
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.
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