Daniel J. Rudolph, Klaus Schmidt
Almost Block Independence and Bernoullicity of $\Bbb Z^d$--Actions by Automorphisms of Compact Abelian Groups
The paper is published: Invent. Math. 120 (1995) 455-488

MSC:

11R56 Adele rings and groups

28D15 General groups of measure-preserving transformations

Abstract: We prove that a $\Bbb Z^d$-action by automorphisms of a compact,
abelian group is Bernoulli if and only if it has completely positive
entropy. The key ingredients of the proof are the extension of certain
notions of asymptotic block independence from $\Bbb Z$-actions to $\Bbb
Z^d$-action and their equivalence with Bernoullicity, and a surprisingly
close link between one of these asymptotic block independence properties
for $\Bbb Z^d$-actions by automorphisms of compact, abelian groups and the
product formula for valuations on global fields.

Keywords: Bernoulli property; $\bbfZ\sp d$-action; block independence