ALMOST BLOCK INDEPENDENCE AND BERNOULLICITY OF ZD-ACTIONS BY AUTOMORPHISMS OF COMPACT ABELIAN-GROUPS

We prove that a 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 Z-actions to Z(d)-action and their equivalence with Bemoullicity, and a surprisingly close link between one of these asymptotic block independence properties for Z(d)-actions by automorphisms of compact, abelian groups and the product formula for valuations on global fields.