[Paper Review] Krieger's finite generator theorem for actions of countable groups III.

Brandon Seward|arXiv (Cornell University)|May 14, 2014

Computability, Logic, AI Algorithms44 references3 citations

TL;DR

This paper extends Rokhlin entropy to non-ergodic actions of countable groups, proving a finite generator theorem that establishes sub-additivity and semi-continuity. It derives formulas using ergodic decompositions and inverse limits, and uses Rokhlin entropy to provide a new proof that ergodic actions with positive sofic entropy have finite stabilizers.

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.

Motivation & Objective

• To extend the finite generator theorem to non-ergodic actions of countable groups using Rokhlin entropy.
• To establish sub-additivity and semi-continuity properties of Rokhlin entropy.
• To derive explicit formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits.
• To clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy.
• To provide a new proof that ergodic actions with positive sofic entropy have finite stabilizers using Rokhlin entropy.

Proposed method

• Adapting the finite generator theorem from ergodic to non-ergodic settings via Rokhlin entropy.
• Using ergodic decomposition to express Rokhlin entropy as an integral over ergodic components.
• Applying inverse limit constructions to analyze the structure of actions and their entropy.
• Leveraging the sub-additivity and semi-continuity of Rokhlin entropy to derive structural constraints.
• Comparing Rokhlin entropy with sofic entropy and classical Kolmogorov–Sinai entropy through measure-theoretic duality.
• Using the finite generator theorem to deduce finiteness of stabilizers in ergodic actions with positive sofic entropy.

Experimental results

Research questions

• RQ1 Can the finite generator theorem be extended to non-ergodic actions of countable groups using Rokhlin entropy?
• RQ2 What are the sub-additivity and semi-continuity properties of Rokhlin entropy in general actions?
• RQ3 How can Rokhlin entropy be expressed via ergodic decompositions and inverse limits?
• RQ4 What is the precise relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy?
• RQ5 Can Rokhlin entropy be used to re-derive the finiteness of stabilizers in ergodic actions with positive sofic entropy?

Key findings

• A non-ergodic finite generator theorem is established, extending the applicability of generator-based entropy analysis to general probability-measure-preserving actions.
• Rokhlin entropy is shown to be sub-additive and semi-continuous with respect to inverse limits and ergodic decompositions.
• Explicit formulas for Rokhlin entropy are derived using ergodic decompositions, expressing it as an integral over ergodic components.
• The paper clarifies that Rokhlin entropy generalizes both sofic entropy and classical Kolmogorov–Sinai entropy in the context of countable group actions.
• Using Rokhlin entropy, a new proof is given that ergodic actions with positive sofic entropy must have finite stabilizers.
• The results demonstrate that Rokhlin entropy provides a unifying framework for understanding entropy invariants across different classes of group actions.