[Paper Review] Finite entropy actions of free groups, rigidity of stabilizers, and a Howe-Moore type phenomenon

Brandon Seward|arXiv (Cornell University)|May 23, 2012

Mathematical Dynamics and Fractals13 references4 citations

TL;DR

This paper investigates finite f-invariant entropy actions of finitely generated free groups, establishing that such actions exhibit strong structural rigidity: stabilizers in factors are either trivial or finite-index, non-identity elements act with infinite Kolmogorov–Sinai entropy when the space is uncountable, and ergodic actions have only finitely many ergodic components for any non-trivial normal subgroup—paralleling a Howe–Moore phenomenon. The results are derived from a new formula for f-invariant entropy and apply to actions with defined, finite entropy.

ABSTRACT

We study a notion of entropy for probability measure preserving actions of finitely generated free groups, called f-invariant entropy, introduced by Lewis Bowen. In the degenerate case, the f-invariant entropy is negative infinity. In this paper we investigate the qualitative consequences of having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that, when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov--Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe--Moore property. Specifically, if the action is ergodic then there is an integer n such that for every non-trivial normal subgroup K the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.

Motivation & Objective

• To understand the qualitative dynamical consequences of finite f-invariant entropy in actions of finitely generated free groups.
• To investigate the structure of stabilizers in factors of such actions, particularly whether they are trivial or finite-index.
• To explore the chaotic behavior of such actions, especially the Kolmogorov–Sinai entropy of non-identity group elements.
• To establish a Howe–Moore-type phenomenon for ergodic actions with finite f-invariant entropy.
• To develop and apply a new formula for f-invariant entropy to derive structural results about the action and its factors.

Proposed method

• The paper introduces a new formula for f-invariant entropy based on conditional entropies of partitions under group actions.
• It uses the generating partition and its iterated images under the group action to compute entropy via a limit involving the ball of radius n in the Cayley graph.
• The key formula involves the weighted sum of Shannon entropies: $ F_G(X, u,S,eta) = (1-2r) ext{H}(eta) + ext{H}(seta \vee \beta) $, where $ r = |S| $, the rank of the free group.
• The authors apply ergodic decomposition techniques to analyze the number of ergodic components of normal subgroups.
• They use measure-theoretic arguments involving atomic measures and entropy bounds to constrain the number of ergodic components.
• A counterexample is constructed using a well-ordering of the group and product measures on coset spaces to show that the Howe–Moore property does not extend to arbitrary subgroups.

Experimental results

Research questions

• RQ1 What are the structural constraints on stabilizers of points in factors of actions with finite f-invariant entropy?
• RQ2 How does finite f-invariant entropy relate to the Kolmogorov–Sinai entropy of individual group elements?
• RQ3 Does finite f-invariant entropy imply a finite number of ergodic components for non-trivial normal subgroups—mirroring the Howe–Moore property?
• RQ4 Can the f-invariant entropy be used to distinguish between different types of group actions, such as Bernoulli shifts or Markov processes?
• RQ5 To what extent does the finite entropy condition restrict the possible dynamics of free group actions on standard probability spaces?

Key findings

• For any factor of an action with finite f-invariant entropy, the stabilizer of almost every point is either trivial or of finite index in the free group.
• In such actions, if the space is not essentially countable, then every non-identity group element acts with infinite Kolmogorov–Sinai entropy.
• For ergodic actions with finite f-invariant entropy, the number of ergodic components of any non-trivial normal subgroup is bounded above by a finite integer depending on the entropy and the generating partition.
• The bound on the number of ergodic components arises from entropy inequalities: $ -(r-1) ext{H}(m) eq f_G(X, u) - ext{H}(eta) $, leading to a finite upper bound on the number of components.
• There exist ergodic actions with finite f-invariant entropy but infinitely many ergodic components for certain infinite cyclic subgroups, showing the bound does not extend to arbitrary subgroups.
• The f-invariant entropy is well-defined and finite if and only if there exists a generating partition with finite Shannon entropy, and the value is independent of the choice of generating set or partition.