Point transitivity, $\Delta$-transitivity and multi-minimality

Abstract

Let $(X, f)$ be a topological dynamical system and $\mathcal {F}$ be a Furstenberg family (a collection of subsets of $\mathbb{N}$ with hereditary upward property). A point $x\in X$ is called an $\mathcal {F}$-transitive point if for every non-empty open subset $U$ of $X$ the entering time set of $x$ into $U$, $\{n\in \mathbb{N}: f^{n}(x) \in U\}$, is in $\mathcal {F}$; the system $(X,f)$ is called $\mathcal {F}$-point transitive if there exists some $\mathcal {F}$-transitive point. In this paper, we first discuss the connection between $\mathcal {F}$-point transitivity and $\mathcal {F}$-transitivity, and show that weakly mixing and strongly mixing systems can be characterized by $\mathcal {F}$-point transitivity, completing results in [Transitive points via Furstenberg family, Topology Appl. 158 (2011), 2221--2231]. We also show that multi-transitivity, $\Delta$-transitivity and multi-minimality can also be characterized by $\mathcal {F}$-point transitivity, answering two questions proposed by Kwietniak and Oprocha [On weak mixing, minimality and weak disjointness of all iterates, Erg. Th. Dynam. Syst., 32 (2012), 1661--1672].