Krieger's Finite Generator Theorem for Ergodic Actions of Countable Groups.

Abstract

For an ergodic probability-measure-preserving action of a countable group G, we define the Rokhlin entropy to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov--Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if the Rokhlin entropy is bounded above by log(k) then there exists a generating partition consisting of k sets. Using this result, we study the properties of Rokhlin entropy as an isomorphism invariant and investigate the still unsolved isomorphism problem for Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that Bernoulli shifts having base spaces of unequal Shannon entropy are non-isomorphic and that Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.