Sensitive dependence on initial conditions and chaotic group 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 ∈ G such that g.U has diameter greater than e. We prove that if a countable G acts transitively on a compact metric space, preserving a probability measure of full support, then the system either is minimal and equicontinuous or has sensitive dependence on initial conditions. Assuming ergodicity, we get the same conclusion without countability. These theorems extend the invertible case of a theorem of Glasner and Weiss. We prove that when a finitely generated, solvable group acts transitively and certain cyclic subactions have dense sets of minimal points, 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 G-actions.