Asymptotic uniform complexity and amenability

TL;DR

This paper introduces the asymptotic uniform complexity as a new invariant for dynamical systems, establishing its relation to amenability and providing criteria for amenability based on this invariant and other properties.

Contribution

It defines the asymptotic uniform complexity, relates it to existing invariants, and characterizes amenability for various classes of dynamical systems and groups.

Findings

Asymptotic uniform complexity equals one for amenable systems.

Established an inequality linking asymptotic uniform complexity and mean topological matching number.

Provided criteria for amenability based on entropy and complexity measures.

Abstract

We introduce a novel quantity for general dynamical systems, which we call the asymptotic uniform complexity. We prove an inequality relating the asymptotic uniform complexity of a dynamical system to its mean topological matching number. Furthermore, we show that the established inequality yields an exact equation for perfect Hausdorff dynamical systems. Utilizing these results, we conclude that a dynamical system is amenable if its asymptotic uniform complexity equals one, and that the converse is true for perfect Hausdorff dynamical systems, such as non-discrete Hausdorff topological groups acting on themselves. Moreover, we establish a characterization of amenability for topologically free dynamical systems on perfect Hausdorff uniform spaces by means of a similar invariant. Furthermore, we provide a sufficient criterion for amenability concerning groups of Lipschitz-automorphisms of precompact pseudo-metric spaces in terms of entropic dimension. Finally, we show that vanishing topological entropy implies the asymptotic uniform complexity to equal one.