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

Abstract

Using the orbital approach to the entropy theory we extend from \mathbb{Z}-actions to general countable amenable group actions T (or provide new short proofs for) the following results: (1) the relative and absolute Krieger Theorem about finite generating partitions (and its infinite Rokhlin counterpart in the case h(T)=\infty); (2) the relative and absolute Sinai Theorem about Bernoullian factors; (3) the Thouvenot Theorem that every intermediate factor of a relatively Bernoullian action is also relatively Bernoullian; (4) the Thouvenot Theorem that a factor of T with the strong Pinsker property enjoys this property; (5) the Smorodinsky–Thouvenot Theorem that T can be spanned by three Bernoullian factors; (6) the Ornstein–Weiss isomorphism theory for Bernoullian actions of the same entropy (provided that they possess generating partitions with at least three elements); and (7) there are uncountably many non-equivalent completely positive entropy extensions of T of the same relative entropy.

In proving these theorems, we were able to bypass the machinery of Orstein and Weiss except for 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).