# Minimality for actions of abelian semigroups on compact spaces with a free interval

## TL;DR

This paper characterizes when abelian semigroup actions on compact spaces with a free interval are minimal and describes the structure of their minimal sets, advancing understanding of dynamical systems with free intervals.

## Contribution

It provides a necessary and sufficient condition for minimal actions and a trichotomy for the structure of minimal sets intersecting a free interval.

## Key findings

- Characterization of minimal actions via necessary and sufficient conditions.
- A trichotomy describing the structure of minimal sets intersecting a free interval.
- Insights into the dynamics of abelian semigroup actions on spaces with free intervals.

## Abstract

We study minimality for continuous actions of abelian semigroups on compact Hausdorff spaces with a free interval. First, we give a necessary and sufficient condition for such a space to admit a minimal action of a given abelian semigroup. Further, for actions of abelian semigroups we provide a trichotomy for the topological structure of minimal sets intersecting a free interval.