Measure-theoretic complexity of ergodic systems

We define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\bar d$$ \end{document}-balls aroundα-n-names necessary to cover most of the system, for any generating partitionα. We show that this rate is essentially bounded if and only if the system is a translation of a compact group, and compute it for several classes of systems of entropy zero, thus getting examples of growth rates inO(n),O(nk) fork ε ℕ, oro(f(n)) for any given unboundedf, and of various relationships with the usual notion of language complexity of the underlying topological system.