Krieger's finite generator theorem for ergodic actions of countable groups I

Brandon Seward

Published 2014-05-14, updated 2015-01-15Version 3

For an ergodic probability-measure-preserving action $G \curvearrowright (X, \mu)$ of a countable group $G$, we define the Rokhlin entropy $h_G^{\mathrm{Rok}}(X, \mu)$ 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 $h_G^{\mathrm{Rok}}(X, \mu) < \log(k)$ then there exists a generating partition consisting of $k$ sets. We actually obtain a notably stronger result which is new even in the case of actions by the integers. Our proofs are entirely self-contained and do not rely on the original Krieger theorem.