Ergodic Theory and Dynamical Systems

Entropy theory without a past

a1 Department of Mathematics, Tel Aviv University, Tel Aviv, Israel (e-mail:
a2 Laboratoire de Probabilités, Université Paris VI, Paris, France
a3 Mathematics Institute, Hebrew University of Jerusalem, Jerusalem, Israel


This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure-preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts the relative disjointness in the sense of Furstenberg, of $0$-entropy extensions from completely positive entropy (c.p.e.) extensions. This theorem is used to prove several classical results in the general setup. For example, we show that the Pinsker factor of a product system is equal to the product of the Pinsker factors of the component systems. Another application is to obtain a generalization (as well as a simpler proof) of the quasifactor theorem for $0$-entropy systems of Glasner and Weiss.

(Received April 15 1998)
(Revised November 6 1998)