Measure-theoretic chaos

Abstract

We define new isomorphism invariants for ergodic measure-preserving systems on standard probability spaces, called measure-theoretic chaos and measure-theoretic+ chaos. These notions are analogs of the topological chaos DC2 and its slightly stronger version (which we denote by DC1 1/2). We prove that: (1) if a topological system is measure-theoretically (measure-theoretically+) chaotic with respect to at least one of its ergodic measures then it is topologically DC2 (DE1 1/2) chaotic; (2) every ergodic system with positive Kolmogorov-Sinai entropy is measure-theoretically+ chaotic (even in a slightly stronger uniform sense). We provide an example showing that the latter statement cannot be reversed, that is, of a system of entropy zero with uniform measure-theoretic+ chaos.