Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits

TL;DR

This paper explores quasi-minimal systems, a weaker form of minimality where dense orbits form an open set, and discusses their properties and relation to minimal and hyperbolic systems.

Contribution

It introduces the concept of quasi-minimality, analyzes its characteristics, and shows that quasi-minimal homeomorphisms on manifolds are not expansive.

Findings

Quasi-minimal systems have dense orbits forming an open set.

Such systems are regarded as parabolic due to elliptic behavior.

Quasi-minimal homeomorphisms on manifolds are not hyperbolic.

Abstract

We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic).