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Dimensional entropy of amenable group actions over stable sets and fibres

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Pith's one-line read Topological conditional entropy of amenable group actions equals the dimensional entropy of stable sets.

desk verdict The paper gives the first topological characterization of conditional entropy via dimensional entropy on stable sets for amenable actions and a purely topological proof of the fibre formula for relative entropy. read the letter →

arxiv 2606.28681 v1 pith:5DPGDN2L submitted 2026-06-27 math.DS

Xinyao He , Guohua Zhang This is my paper · ORCID
classification math.DS
keywords dimensionalentropyamenablegroupactionsstablesetsfibrestopologicalconditionalrelativefactormaps
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that Bowen's dimensional entropy computed on stable sets recovers the topological conditional entropy exactly for actions of amenable groups. It also proves that the relative topological entropy of any factor map equals the dimensional entropy of the fibers over points in the base space. Both results are obtained with purely topological arguments that avoid measures. The characterizations extend single-map results to group actions and supply the first such description of conditional entropy via dimensional entropy even when the acting group is the integers.

What carries the argument

Bowen's dimensional entropy defined on stable sets and on fibers of factor maps for amenable group actions, used to equate classical entropy quantities to covering-based quantities on those sets.

What would settle it

An explicit amenable group action together with a concrete stable set for which the numerical value of Bowen's dimensional entropy differs from the value of the topological conditional entropy.

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Extended reading notes

Core claim

Bowen's dimensional entropy on the stable sets of an amenable group action equals the topological conditional entropy of the action, and the dimensional entropy on the fibers of a factor map equals the relative topological entropy of that map. These identities hold for the full class of amenable groups and are proved without invoking invariant measures or Shannon-McMillan-Breiman theorems.

Load-bearing premise

The definitions and covering properties of Bowen's dimensional entropy on stable sets and fibers extend directly to amenable group actions so that the stated equalities hold.

Editorial extensions

If this is right

  • Topological conditional entropy can be recovered by computing dimensional entropy only on stable sets rather than on the whole space.
  • Relative topological entropy of a factor map equals the supremum of dimensional entropies of its fibers.
  • A dimensional-entropy inequality holds for any factor map relating the entropy of a set, its image, and the entropies of the fibers.
  • The fiber characterization of relative entropy admits a purely topological proof that applies to all amenable groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The stable-set formula may simplify explicit calculations of conditional entropy in systems whose stable sets admit simple covers.
  • The same covering arguments could be tested on actions of non-amenable groups once a suitable definition of dimensional entropy is chosen.
  • The fiber formula supplies a way to compare relative entropies across different factor maps by examining only the preimage sets.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper extends Bowen's dimensional entropy to actions of amenable groups using Følner sequences. It proves three main results: Theorem 1.1 characterizes topological conditional entropy via the dimensional entropy of stable sets (answering a question from Dou-Wang-Zhang 2025 and providing the first such result even for Z-actions); Theorem 1.2 gives a dimensional entropy inequality for factor maps relating the entropy of a set, its image, and the topological entropy of fibres; and Theorem 1.3 shows that the relative topological entropy of a factor map equals the dimensional entropy of the fibres, via a purely topological argument generalizing Oprocha-Zhang 2011 and contrasting with measure-theoretic approaches.

Significance. If the results hold, they strengthen the topological foundations of entropy theory for amenable group actions by supplying new characterizations and inequalities that avoid measure-theoretic tools, generalize prior work from single maps to group actions, and resolve an open question on conditional entropy. The explicit use of Følner sequences for the extension and the self-contained topological proofs are notable strengths.

minor comments (2)
  1. [Abstract / §1] The abstract and introduction could include a short explicit statement of the precise definition of Bowen's dimensional entropy on stable sets (e.g., via the Følner-sequence limit) to make the extension from the Z-case immediately visible without consulting the cited references.
  2. [§1] Theorem 1.2 is described as the dimensional-entropy counterpart of the packing-entropy result in Dou-Zheng-Zhou 2023; a one-sentence comparison of the two inequalities (e.g., noting where the dimensional version replaces packing with Bowen dimension) would clarify the novelty for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the referee recognizes the novelty of the characterizations and the purely topological proofs.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The manuscript supplies explicit definitions of Bowen's dimensional entropy extended to amenable actions via Følner sequences, then derives the characterizations in Theorems 1.1–1.3 by direct topological arguments. Prior results (Dou-Wang-Zhang 2025, Dou-Zheng-Zhou 2023, Oprocha-Zhang 2011) are cited only for context and motivation; the proofs here are stated to be independent and self-contained, with no reduction of any claimed equality or inequality to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The central claims therefore stand on their own derivations rather than on inputs by construction.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Paper relies on standard background in topological dynamics and amenable group theory; no free parameters or invented entities introduced in the abstract.

assumptions (2)
  • domain assumption Amenable groups admit Følner sequences allowing well-defined entropy limits.
    Invoked throughout for extending single-map results to group actions (Theorems 1.1-1.3).
  • domain assumption Bowen's dimensional entropy is well-defined on stable sets and fibres for the given actions.
    Central to all three theorems; assumed to behave as in prior single-map cases.

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Cite this review

Pith. "Pith review of Dimensional entropy of amenable group actions over stable sets and fibres." pith.science (2026). https://pith.science/paper/5DPGDN2L

@misc{pith2026260628681,
  author       = {Pith},
  title        = {Pith review of: Dimensional entropy of amenable group actions over stable sets and fibres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5DPGDN2L}},
  note         = {Machine review of arXiv:2606.28681}
}
abstract

This paper is devoted to the study of Bowen's dimensional entropy on subsets for actions of amenable groups. We prove three main results. (1) First, topological conditional entropy is characterized by the dimensional entropy of stable sets (Theorem 1.1), answering a question of Dou, Wang and the second author of the present paper raised in [Fund. Math., 2025]. We remark that our Theorem 1.1 is the first characterization of topological conditional entropy via Bowen's dimensional entropy of stable sets even for $\mathbb{Z}$-actions. (2) Second, we establish a dimensional entropy inequality for factor maps (Theorem 1.2). It relates dimensional entropy of a set to that of its image and topological entropy of fibres, and may be viewed as the dimensional-entropy counterpart of the factor-map inequality for packing topological entropy due to Dou, Zheng, and Zhou proved as Theorem 1.4 in [Ergodic Theory Dynam. Systems, 2023]. (3) Third, the relative topological entropy of a factor map is determined by the dimensional entropy of the fibres (Theorem 1.3). Notably, our proof of this formula (Theorem 1.3) is purely topological, in contrast to the recent measure-theoretic approach of Dou, Wang and Zhou based on relative Shannon--McMillan--Breiman theorems. These results (Theorem 1.2 and 1.3) not only generalize the work of Oprocha and the second author of the present paper [Nonlinearity, 2011] from single transformations to amenable group actions, but also provide a purely topological and self-contained proof of a fibre entropy characterization recently obtained through measure-theoretic arguments.

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