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].

19 pages, the final version. Change the title to "Point transitivity, $\Delta$-transitivity and multi-minimality"

References in corpus (2)

• On weak mixing, minimality and weak disjointness of all iterates
• Transitive points via Furstenberg family

Cited by in corpus (2)

• Positive topological entropy and $\Delta$-weakly mixing sets
• Furstenberg families and transitivity in non-autonomous systems