Ergodic Theory and Dynamical Systems

Research Article

$\mathbb {Z}^d$ group shifts and Bernoulli factors

MIKE BOYLEa1a2 and MICHAEL SCHRAUDNERa2

a1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA (email: mmb@math.umd.edu)

a2 Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile, Chile (email: mschraudner@dim.uchile.cl)

Abstract

In this paper, a group shift is an expansive action of $\Z ^d$ on a compact metrizable zero-dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equal-entropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraically equivalent to these Bernoulli factors. A completely positive entropy group (even vector) shift need not be topologically conjugate to a Bernoulli shift, and the Pinsker factor of a vector shift need not split topologically.

(Received May 09 2007)

(Revised August 28 2007)