Sensitive dependence on initial conditions and chaotic group actions

TL;DR

This paper explores how group actions on compact spaces exhibit sensitive dependence on initial conditions, linking measure preservation, minimality, and equicontinuity, and provides new examples of complex group actions.

Contribution

It generalizes a theorem of Glasner and Weiss to non-invertible actions and constructs novel examples of non-compact group actions with specific dynamical properties.

Findings

Systems with measure-preserving actions are either minimal and equicontinuous or sensitive.

Finitely generated solvable group actions with certain cyclic subactions are sensitive.

Constructed examples include non-compact monothetic groups with transitive, non-minimal actions.

Abstract

A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number e>0 such that for any open set U we can find g in G such that g.U has diameter greater than e. We prove that if a G action preserves a probability measure of full support, then the system is either minimal and equicontinuous, or has sensitive dependence on initial conditions. This generalizes the invertible case of a theorem of Glasner and Weiss. We prove that when a finitely generated, solvable group, acts and certain cyclic subactions have dense sets of minimal points, then the system has sensitive dependence on initial conditions. Additionally, we show how to construct examples of non-compact monothetic groups, and transitive, non-minimal, almost-equicontinuous, recurrent, group actions.