Minimality and Gluing Orbit Property

TL;DR

This paper investigates the relationship between minimality, topological entropy, and the gluing orbit property in dynamical systems, highlighting conditions under which these properties are equivalent or distinct.

Contribution

It establishes that systems are either minimal or have positive entropy and shows equivalence of transitivity, minimality, and orbit gluing in equicontinuous systems.

Findings

Systems are either minimal or have positive topological entropy.

In equicontinuous systems, transitivity, minimality, and orbit gluing are equivalent.

The paper clarifies the relationship between gluing orbit property and specification property.

Abstract

We show that a topological dynamical system is either minimal or have positive topological entropy. Moreover, for equicontinuous systems, we show that topological transitivity, minimality and orbit gluing property are equivalent. These facts reflect the similarity and dissimilarity of gluing orbit property with specification property.