The systems with almost Banach mean equicontinuity for abelian group actions

Bin Zhu; Xiaojun Huang; Yuan Lian

REVIEW 2 major objections 5 minor 25 references

The systems with almost Banach mean equicontinuity for abelian group actions

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For a transitive abelian group action, a single almost Banach-mean equicontinuous point forces zero topological entropy.

desk verdict A worthwhile equivalence theorem and a coherent no-isolated-points entropy proof, but the headline zero-entropy theorem is false as stated because of a simple isolated-point counterexample. read the letter →

arxiv 1909.00920 v1 pith:HT7JH223 submitted 2019-09-03 math.DS

Bin Zhu , Xiaojun Huang , Yuan Lian This is my paper · ORCID

classification math.DS MSC 37A3537B40

keywords Banach-meanequicontinuityWeyl-meanBesicovitch-meantopologicalentropyIE-pairindependencedensityabeliangroupactionmeansensitivity

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

This paper introduces Banach-mean equicontinuity for group actions on compact metric spaces, defined through the upper Banach density of the times at which two nearby orbits remain close, and proves that for countable abelian group actions this notion coincides with Weyl-mean and Besicovitch-mean equicontinuity. Its main theorem states that if a countably infinite abelian group acts transitively and the action has at least one Banach-mean equicontinuous point, then the topological entropy of the action is zero. The proof establishes the contrapositive through a dichotomy: a transitive action is either almost Weyl-mean equicontinuous or Weyl-mean sensitive, and positive entropy forces the latter through an independence-pair argument inside the independence-density framework. As a corollary, every Banach-mean equicontinuous action of a countable abelian group on a compact metric space has zero entropy. The result matters because it carries the zero-entropy consequences of mean equicontinuity, previously developed for integer actions, to all countable abelian group actions using combinatorial independence rather than invariant measures.

What carries the argument

The central object is the Banach-mean pseudometric $D(x,y)=\inf_{F\in\operatorname{Fin}(G)}\sup_{g\in G}\frac{1}{|F|}\sum_{t\in Fg} d(tx,ty)$, which measures how far two orbits diverge when averaged over right translates of a finite window; the proof uses the theorem that for amenable groups this equals the Weyl-mean pseudometric $D(x,y)$. The combinatorial engine is the IE-pair: a pair of distinct points $x_1,x_2$ such that every product neighbourhood $(U_1,U_2)$ has positive independence density, together with the theorem that positive topological entropy implies the existence of an IE-pair. Given an IE-pair, the argument finds a recurrent transitive point $x_0$, picks two return times $l_1,l_2$ whose shifted independence sets remain large in Banach density, and forces $l_1x_0$ and $l_2x_0$ to be close in the metric but far apart in the Weyl pseudometric, contradicting almost Weyl-mean equicontinuity. This turns positive entropy into Weyl-mean sensitivity.

What would settle it

Build a compact metric space with an isolated point carrying a transitive action of a countably infinite abelian group that is almost Banach-mean equicontinuous and has positive topological entropy; such a system would refute Theorem 1.1 as stated. If instead every transitive abelian group action on a space with isolated points has zero entropy, then the missing case closes and the statement survives.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: let $G$ be a countably infinite abelian group, $X$ a compact metric space, and $G \curvearrowright X$ a transitive continuous action. If the action is almost Banach-mean equicontinuous, meaning some point $x$ has the property that for every $\varepsilon>0$ there is $\delta>0$ such that every $y$ within $\delta$ of $x$ satisfies the Banach-mean bound $D(x,y)<\varepsilon$, then $h_{\mathrm{top}}(X,G)=0$. The proof runs through Theorem 6.6, which shows the contrapositive for compact metric spaces without isolated points: positive topological entropy implies the action is Weyl-mean sensitive. Theorem 4.3 identifies the Banach-mean and Weyl-mean pseudometrics for amenable groups, so the almost equicontinuity assumption places the system on the equicontinuous side of the dichotomy in Theorem 5.5, leaving no room for positive entropy.

Load-bearing premise

The proof needs a starting point whose orbit is dense and which returns infinitely often to every neighbourhood of itself; the paper obtains such a point only when the space has no isolated points, a condition Theorem 1.1 does not state.

Editorial extensions

If this is right

A transitive action of a countably infinite abelian group on a compact metric space that is almost Banach-mean equicontinuous has zero topological entropy (Theorem 1.1).
For countable abelian group actions, Banach-, Weyl-, and Besicovitch-$F$-mean equicontinuity are the same property, so each of them forces zero entropy in the transitive setting.
Every Banach-mean equicontinuous action of a countable abelian group on a compact metric space, transitive or not, has zero topological entropy (Theorem 1.2).
A transitive action of a countable amenable group is either almost Weyl-mean equicontinuous or Weyl-mean sensitive; positive entropy puts it in the sensitive branch whenever the space has no isolated points (Theorems 5.5 and 6.6).

Reading between the lines

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

Because the dichotomy and the Banach/Weyl pseudometric equality rely only on amenability, the same zero-entropy argument may extend to countable amenable groups beyond the abelian case, provided the Besicovitch/Weyl equivalence and the IE-pair machinery hold there; the paper proves the equivalence only for abelian groups.
A testable extension is whether almost Banach-mean equicontinuity without transitivity already forces zero entropy; the paper's Theorem 1.2 covers only the fully equicontinuous case in the non-transitive setting, leaving the almost case open.
Settling Theorem 1.1 for spaces with isolated points would close the gap between the theorem's statement and Theorem 6.6's no-isolated-points hypothesis; a transitive system with an isolated point and positive entropy would disprove the theorem as stated.
The independence-density mechanism—an IE-pair yielding two disjoint shifted independence sets that separate nearby points in the mean pseudometric—could be adapted to other sensitivity notions, such as Besicovitch sensitivity, wherever a Furstenberg correspondence principle is available.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper proves something true and interesting—Banach-mean equicontinuity collapses into Weyl- and Besicovitch-mean equicontinuity for abelian group actions—but the headline theorem as stated is false. There is a simple counterexample with an isolated point. That is not a minor gap.

The new notion (Definition 4.1) is a natural Banach-density analog of existing mean metrics. The proof of Theorem 4.3 that the Banach pseudometric D equals the Weyl pseudometric D looks correct: the Følner-shift argument is standard, and the contradiction step is sound. Theorem 4.5, which adds Besicovitch into the equivalence, is essentially quoted from Fuhrmann–Gröger–Lenz [11]; the paper should say so more explicitly, but the dependency is legitimate. The IE-pair route to zero entropy in Theorem 6.6 is also coherent: choose a transitive recurrent point, use the independence set of an IE-pair, and produce two points in a small ball with Weyl-mean distance bounded away from zero. Under the explicit assumption that X has no isolated points, that argument works.

The problem is Theorem 1.1. The abstract and introduction state it for every compact metric transitive system. The proof imports Theorem 6.6, which assumes no isolated points. The stress-test counterexample is exactly what one would fear: take a transitive point in the full 2-shift, insert a single star at time 0, and take the orbit closure. The star point is isolated, hence trivially Banach-mean equicontinuous, while the closure contains the entire 2-shift, so entropy is log 2. So Theorem 1.1 is not merely unproven in the isolated-points case; it is false. The paper already contains the right fix—state the theorem for systems without isolated points—but the current wording is a genuine mathematical error.

There are smaller issues: Proposition 5.3 assumes points with D(u,v)>2δ to conclude Weyl-mean sensitivity with δ; the δ/2δ mismatch should be cleaned up. The abstract has a typo ('Bnanach'). These are trivial. The citation pattern is healthy; the paper builds on Kerr–Li and Li–Tu–Ye without obvious overselling.

Who should read this: people working on mean equicontinuity and the structure of abelian actions will want the equivalence result and the corrected zero-entropy theorem. The paper is not publishable as is, but a revision that restricts Theorem 1.1 to no-isolated-points and adjusts the abstract and introduction would make it a useful contribution. I would send it to a referee; the core mathematics is worth checking carefully, and the counterexample to the stated theorem is the kind of thing a referee is supposed to catch.

Referee Report

2 major / 5 minor

Summary. The paper defines a Banach-mean pseudometric D for actions of countable discrete groups on compact metric spaces and studies the corresponding equicontinuity notions. For countable amenable group actions it proves D equals the Weyl-mean pseudometric (Theorem 4.3), and for abelian group actions it asserts the equivalence of Banach-, Weyl-, and Besicovitch-mean equicontinuity (Theorem 4.5). The main advertised result is Theorem 1.1: every transitive, almost Banach-mean equicontinuous action of a countably infinite abelian group has zero topological entropy. Section 6 contains the proof via IE-pairs, Følner independence, and a dichotomy between almost Weyl-mean equicontinuity and Weyl-mean sensitivity; Theorem 1.2 derives the zero-entropy conclusion for globally Banach-mean equicontinuous actions using the variational principle.

Significance. If the results were correct in the stated generality, the paper would give a useful extension of mean-equicontinuity/zero-entropy results from Z-actions to abelian group actions. The equality D=D̄ in Theorem 4.3 is a clean observation, and the IE-pair argument in Theorem 6.6 is nontrivial and appears largely coherent for systems without isolated points. However, Theorem 1.1 is false as stated: the proof requires a no-isolated-points hypothesis, and a simple counterexample with an isolated equicontinuous point and positive entropy shows the omission is essential. The headline claim is therefore not valid.

major comments (2)

[Theorem 1.1 (Abstract and Introduction); Theorem 6.6] Theorem 1.1 is false as stated because the proof goes through Theorem 6.6, which assumes X has no isolated points, and that hypothesis is essential. Let A={0,1,*}, choose y∈{0,1}^Z with dense forward orbit in the full two-shift, and define x∈A^Z by x_0=* and x_n=y_n for n≠0. Let X be the closure of {σ^n x: n∈Z}. The Z-action on X is transitive (x is a transitive point). The cylinder [* at 0] meets X only in {x}, so x is isolated; hence for δ small enough B(x,δ)={x} and D(x,y)=0<ε for every y∈B(x,δ), making x a Banach-mean equicontinuous point. Thus (X,Z) is almost Banach-mean equicontinuous. On the other hand, for every w∈{0,1}^Z there is a sequence n_k→∞ with σ^{n_k}y→w, and then σ^{n_k}x→w, so X contains the full two-shift as a closed invariant subset and h_top(X,Z)≥log2>0. This contradicts Theorem 1.1 exactly in the omitted isolated-points case.
[Proposition 5.3] The proof begins with the stronger hypothesis that every open set U contains u,v with D(u,v)>2δ, whereas the proposition is stated with >δ. As written the proof does not establish the stated implication. The gap is readily fixable: the stated hypothesis with constant δ gives Weyl-mean sensitivity with sensitivity constant δ/2 by the same triangle inequality used in the proof. Still, the mismatch must be corrected because Proposition 5.3 is used in the proof of Theorem 5.5 and hence in Theorem 6.6.

minor comments (5)

[Definition 2.1] The definition of a transitive point as Gx=X should read overline{Gx}=X (dense orbit), consistent with Proposition 2.2 and with the later usage in the paper.
[Abstract and Definitions 4.1, 4.2] There are typos in the abstract and introduction: 'Bnanach', 'Wely', 'abelain', and 'alelian'; also 'for for every' appears in Definition 4.1 and Definition 4.2.
[Theorem 4.5] The invoked theorem [11, Theorem 1.3, p.6] should be stated explicitly, since Theorem 4.5 is one of the paper's main equivalences and the reader cannot otherwise verify it.
[Section 6, proof of Theorem 6.6] After proving J1∩J2=∅, the text says 'Hence, J1∩J2 ⁄= ∅'; this should be '=∅'.
[Section 6, proof of Theorem 1.1] The one-line proof of Theorem 1.1 should be expanded to name the results that convert almost Banach-mean equicontinuity to almost Weyl-mean equicontinuity (Theorem 4.3/Corollary 4.4) and to flag the no-isolated-points hypothesis from Theorem 6.6.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is derived from external IE-pair and mean-equicontinuity machinery, not from its own conclusion.

full rationale

The paper's new Banach-mean notion is defined independently in Definition 4.1 and is then shown to coincide with the Weyl-mean pseudometric in Theorem 4.3 by a genuine Følner-set argument; this is a proved equivalence, not a definitional identification. The main result (Theorem 1.1) is proved by contrapositive through Theorem 6.6, which relies on the external Kerr–Li IE-pair theorem (Theorem 3.9, cited from [20, Theorem 12.19]), a Følner independence-density argument (Proposition 6.4 and Lemma 6.5), and standard recurrence facts (Proposition 2.4). No quantity in the conclusion is fitted from, or defined in terms of, the entropy value, and no load-bearing self-citation chain forces the conclusion. The only serious defect is a correctness gap: Theorem 6.6 explicitly assumes X has no isolated points, while the proof of Theorem 1.1 in Section 6 does not remove that assumption, so the theorem as stated in the abstract is not proved for systems with isolated points. That is a mathematical gap or false-statement issue, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard theorems from amenable group dynamics and one cited Foelner-independence theorem. No empirical parameters are fitted. The only load-bearing condition not stated in the main theorem is the no-isolated-points assumption imported from Theorem 6.6.

assumptions (6)

standard math Every abelian group is amenable (Theorem 3.2, cited from Ceccherini-Silberstein and Coornaert).
Used to apply Foelner-sequence constructions and the variational principle.
standard math IE2(X,G) \ Delta2(X) is nonempty if and only if htop(X,G) > 0 (Theorem 3.9, cited from Kerr-Li).
Bridges positive entropy to independence pairs in Theorem 6.6.
standard math For abelian group actions, Besicovitch-F-mean equicontinuity is independent of the Foelner sequence F (Theorem 1.3 of Fuhrmann-Groeger-Lenz).
Used in Theorem 4.5 to identify Besicovitch-, Weyl- and Banach-mean equicontinuity.
standard math In a transitive system on a compact metric space without isolated points, recurrent points form a dense G_delta (Proposition 2.4, cited from Kerr-Li).
Provides the point x0 in Tran(X,G) intersect Re(X,G) used in Theorem 6.6.
standard math Variational principle for amenable group actions (Theorem 7.5, cited from Huang-Ye-Zhang).
Used in Theorem 1.2 to lift measure entropy zero to topological entropy zero.
standard math Zorn's lemma, used to select the maximal set H in Theorem 6.6.
Standard set-theoretic tool, harmless.

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Cite this review

Pith. "Pith review of The systems with almost Banach mean equicontinuity for abelian group actions." pith.science (2026). https://pith.science/paper/HT7JH223

@misc{pith2026190900920,
  author       = {Pith},
  title        = {Pith review of: The systems with almost Banach mean equicontinuity for abelian group actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HT7JH223}},
  note         = {Machine review of arXiv:1909.00920}
}

read the original abstract

In this paper, we give the concept of Banach-mean equicontinuity and prove that three concepts, Bnanach-, Weyl- and Besicovitch-mean equicontinuity of a dynamic system with abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system with abelain group action is zero. As an application with our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an abelian groups is zero.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

The structure of mean equicontinuous group actions

G. Fuhrmann, M. Gr¨ oger, and D. Lenz, The structure of mean equacontinuous group actions. arXiv preprint. Available at: https://arxiv.org/pdf/1812.10219v1

work page Pith review arXiv

[2] [1]

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work page 1959

[3] [2]

Auslander, Minimal Flows and Their Extensions

J. Auslander, Minimal Flows and Their Extensions. North-holland, 1988

work page 1988

[4] [3]

E. Akin, J. Auslander and K. Berg. when is a transitive map chaotic? Convergence in Ergodic Theory and Prob- ability(Conlumbus,OH,199 3)(Ohio State university Mathe matic Research Institute Publication,5).de Gruyter, Berlin,1996,pp.25-40

work page 1996

[5] [4]

Mathias Beiglb¨ ock, Vitaly Bergelson, and Alexander Fish, Sumset phenomenon in countable amenable groups. Adv. Math. 223 (2010), 416-432

work page 2010

[6] [5]

Baake, D

M. Baake, D. Lenz, and R. V. Moody, Characterization of model sets by dynamical systems. Ergos. Th. & Dynam. Sys. 27(2) (2007), 341-382

work page 2007

[7] [6]

Ceccherini-Silberstein, M

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Denker, C

M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol. 527, Springer, Berlin, 1976

work page 1976