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

2017/05/26 by Andrei Alpeev, Alpeev, Andrei, Brandon Seward +1 · 1 citation
Computer Science · Mathematics · #37A15 #37A35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1705.09707

openalex publication_date 2017/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov--Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

Cited by

• Surjunctivity does not characterize cosoficity of invariant random subgroups

Related

• Krieger's finite generator theorem for actions of countable groups II
• Krieger's finite generator theorem for actions of countable groups I
• Positive entropy actions of countable groups factor onto Bernoulli\n shifts
• Finite entropy actions of free groups, rigidity of stabilizers, and a Howe-Moore type phenomenon
• The Shannon-McMillan-Breiman theorem beyond amenable groups