  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.
