[Paper Review] Bernoulli actions and infinite entropy

David Kerr, Hanfeng Li|arXiv (Cornell University)|May 27, 2010

Computability, Logic, AI Algorithms4 references3 citations

TL;DR

This paper establishes that for countable sofic groups, a Bernoulli action with an infinite entropy base has infinite sofic measure entropy with respect to every sofic approximation sequence. Using a topological representation of dynamics and entropy lower bounds, the authors prove that the entropy of such Bernoulli actions equals the base entropy independently of the approximation sequence, resolving a question posed by Bowen and extending classical entropy theory to non-amenable and non-Ornstein sofic groups.

ABSTRACT

We show that, for countable sofic groups, a Bernoulli action with infinite entropy base has infinite entropy with respect to every sofic approximation sequence. This builds on the work of Lewis Bowen in the case of finite entropy base and completes the computation of measure entropy for Bernoulli actions over countable sofic groups. One consequence is that such a Bernoulli action fails to have a generating countable partition with finite entropy if the base has infinite entropy, which in the amenable case is well known and in the case that the acting group contains the free group on two generators was established by Bowen using a different argument.

Motivation & Objective

• To resolve a question posed by Lewis Bowen regarding the behavior of sofic measure entropy in Bernoulli actions with infinite entropy base.
• To extend the entropy classification of Bernoulli actions beyond amenable groups and groups containing F₂ to all countable sofic groups.
• To show that such actions do not admit a generating measurable partition with finite entropy, generalizing known results in the amenable and F₂-containing cases.
• To complete the computation of sofic measure entropy for all Bernoulli actions over countable sofic groups by proving entropy equals base entropy universally.

Proposed method

• Represent the dynamics in a topological framework using continuous actions on compact metrizable spaces and unital *-homomorphisms into C*(d).
• Define a pseudometric on homomorphisms using finite partitions of unity in C(X), enabling approximation of measure-theoretic entropy via topological entropy techniques.
• Use Stirling’s approximation and entropy bounds to estimate the number of approximately equivariant, approximately measure-preserving maps from L∞(X,μ) to C^d.
• Establish a lower bound on the exponential growth rate of ε-separated sets in the space of such homomorphisms, using combinatorial entropy estimates and uniform continuity of the entropy function ξ(t) = -t log t.
• Apply a variant of Bowen’s finite entropy lower bound argument in the infinite entropy regime, leveraging the structure of sofic approximations and measure concentration.
• Combine the lower bound with the known finite entropy case to show that infinite base entropy implies infinite sofic measure entropy for all approximation sequences.

Experimental results

Research questions

• RQ1 Does a Bernoulli action with infinite base entropy over a countable sofic group have infinite sofic measure entropy for every sofic approximation sequence?
• RQ2 Can the entropy of a Bernoulli action over a countable sofic group be computed uniformly as the base entropy, regardless of the approximation sequence?
• RQ3 Does the absence of a generating finite-entropy partition in Bernoulli actions with infinite base entropy hold for all countable sofic groups, not just amenable or F₂-containing ones?
• RQ4 Is the conjugacy class of a Bernoulli action over a countable sofic group determined solely by the base entropy, even when the base has infinite entropy?

Key findings

• For any countable sofic group G and sofic approximation sequence Σ, the sofic measure entropy of a Bernoulli action with base (X,μ) is equal to H(μ), the base entropy, regardless of whether H(μ) is finite or infinite.
• When H(μ) = +∞, the sofic measure entropy hΣ,μ^G(X^G, G) is also +∞ for every sofic approximation sequence Σ.
• There is no generating countable measurable partition Q of X^G such that H_μ^G(Q) < ∞ when H(μ) = +∞, which generalizes known results in the amenable and F₂-containing cases.
• The result completes the entropy classification of Bernoulli actions over countable sofic groups, showing that conjugate Bernoulli actions must have the same base entropy.
• The proof establishes that infinite entropy is preserved across all sofic approximations, using topological and combinatorial entropy estimates based on Stirling’s formula and measure concentration.
• The work resolves a question of Weiss and extends Bowen’s earlier results to all countable sofic groups, including non-amenable, torsion groups that do not contain F₂.