Transitive points via Furstenberg family

TL;DR

This paper classifies transitive dynamical systems using Furstenberg families, linking various types of mixing and transitivity properties to specific families, and explores their implications and examples.

Contribution

It introduces a classification of transitive systems via Furstenberg families, establishing new characterizations of weak mixing and other properties, and constructs examples distinguishing these classes.

Findings

Weakly mixing systems are $_{ip}$-point transitive.

Every transitive system with dense small periodic sets is disjoint from totally minimal systems.

A system is $ riangle^*(_{wt})$-transitive iff it is weakly disjoint from all P-systems.

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 $\{\textrm{D-sets}\}$-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.