Descriptive Estimates for a Set of Points That Approximate an Ergodic Measure

Abstract

We obtain descriptive estimates for a set of points that approximate an ergodic invariant measure of a continuous mapping on a compact set. For example, in the case of a metrically transitive mapping with an invariant measure equivalent to the Lebesgue measure, we prove that a set of points generating invariant measures with maximum support contains a dense G _δ-set, whereas, in the general case, one has a much worse estimate G _δσδ