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

TL;DR

This paper extends Krieger's finite generator theorem to all countably infinite groups by introducing Rokhlin entropy, showing that low entropy actions admit finite generating partitions, thus generalizing classical entropy results.

Contribution

The paper proves Krieger's finite generator theorem for all countably infinite groups using the concept of Rokhlin entropy, broadening the scope beyond amenable groups.

Findings

Rokhlin entropy is a natural analogue of classical entropy for group actions.

For actions with Rokhlin entropy less than log(k), a k-set generating partition exists.

The theorem applies to all countably infinite groups, not just amenable ones.

Abstract

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.