On subsets of integers having dense orbits

TL;DR

This paper investigates the properties of subsets of natural numbers related to dense orbits in minimal dynamical systems, providing characterizations and answering longstanding questions about their behavior across different system classes.

Contribution

It characterizes transitive systems disjoint from totally minimal systems and links scattering properties to the $R$-sequence behavior, addressing open questions in dynamical systems theory.

Findings

A system is scattering if and only if certain orbit conditions hold.

If scattering and weak scattering differ, then specific open questions have negative answers.

Provides a characterization of transitive systems disjoint from all totally minimal systems.

Abstract

Let $A\subset \mathbb{N}$. We say $A$ is an $R$-sequence for a given minimal system $(Y,S)$ if there is $y\in Y$ such that $\{S^ny:n\in A\}$ is dense in $Y$. Richter asked if $A$ is an $R$-sequence for all minimal equicontinuous systems implies that $A$ is an $R$-sequence for all minimal systems. In this paper, we investigate this question and related issues within the framework of totally minimal systems, including a characterization of transitive systems that are disjoint from all totally minimal systems. A dynamical system is scattering (resp. weakly scattering) if its product with any minimal (resp. minimal and equicontinuous) system is transitive. It turns out that $(X,T)$ is scattering if and only if for any transitive point $x\in X$ and any minimal system $(Y,S)$ there is $y\in Y$ such that the orbit of $(x,y)$ is dense in $X\times Y$ if and only if for each transitive point $x\in X$ and any non-empty open subset $U$ of $X$, $\{n\in \mathbb{N}:T^nx\in U\}$ is an $R$-sequence. By combining this result with earlier work of Huang and Ye, we deduce that if scattering and weak scattering are distinct properties, then both Richter's question and Katznelson's question admit negative answers.