Generators and Bernoullian factors for amenable actions and cocycles on their orbits

Abstract

Using the orbital approach to the entropy theory we extend from Zactions to general countable amenable group actions T (or provide new short proofs to) the following results: (1) relative and absolute Krieger theorem about finite generating partitions (and its infinite Rokhlin counterpart in case h(T ) = ∞), (2) relative and absolute Sinai theorem about Bernoullian factors, (3) Thouvenot theorem that every intermediate factor of a relatively Bernoullian action is also relatively Bernoullian, (4) Thouvenot theorem that a factor of T with the strong Pinsker property enjoys this property, (5) Smorodinsky-Thouvenot theorem that T can be spanned by three Bernoullian factors, (6) Ornstein-Weiss isomorphism theory for Bernoullian actions of the same entropy (provided that they possess generating partitions with at least 3 elements), (7) there are uncountably many non-equivalent CPE extensions of T of the same relative entropy, etc. In proving these theorems, we were able to bypass the machinery from [OrW] except of the Rokhlin lemma. It is shown that the language of orbit equivalence relations and their cocycles (unlike the standard dynamical one) is well suited for the inducing operation needed to settle (1), (5) and (6).