Asymptotic uniform complexity and amenability

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.