MEASURE-THEORETIC COMPLEXITY OF ERGODIC SYSTEMS

We define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate in n of the number of small (d) over bar-b alls around alpha-n-names necessary to cover most of the system, for a ny generating partition alpha. We show that this rate is essentially b ounded 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 ge tting examples of growth rates in O(n), O(n(k)) for k is an element of N, or o(f(n)) for any given unbounded f, and of various relationships with the usual notion of language complexity of the underlying topolo gical system.