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

TL;DR

This paper explores the concept of $$-point transitivity in topological dynamical systems, linking it to various mixing and minimality properties, and provides characterizations that answer open questions in the field.

Contribution

It establishes new characterizations of weakly mixing, strongly mixing, multi-transitivity, $ riangle$-transitivity, and multi-minimality using $$-point transitivity, extending previous results.

Findings

Characterizes weakly and strongly mixing systems via $$-point transitivity.

Provides $$-point transitivity characterizations for multi-transitivity and $ riangle$-transitivity.

Answers open questions on the relationship between these properties and $$-point transitivity.

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].