Chaotic almost minimal actions

TL;DR

This paper introduces a new class of dynamical systems called chaotic almost minimal actions, which exhibit chaos while being close to minimal, expanding understanding of invariant sets under group actions.

Contribution

It defines chaotic almost minimal actions, explores conditions for their existence, and demonstrates examples for both  and ^d groups with multiple ergodic measures.

Findings

Existence of chaotic almost minimal -action.

Existence of ^d-actions with multiple ergodic measures.

Characterization of groups admitting such actions.

Abstract

Motivated by Furstenberg's Theorem on sets in the circle invariant under multiplication by a non-lacunary semigroup, we define a general class of dynamical systems possessing similar topological dynamical properties. We call such systems chaotic almost minimal, reflecting that these systems are chaotic, but in some sense are close to minimal. We study properties of the acting group needed to admit such an action, and show the existence of a chaotic almost minimal $\mathbb{Z}$-action. We show there exists chaotic almost minimal $\mathbb{Z}^{d}$-actions which support multiple distinct nonatomic ergodic probability measures.