Topological dynamics for the endograph metric I: Equivalences with other metrics

Antoni L\'opez-Mart\'inez

REVIEW 1 major objections 4 minor 2 cited by

Topological dynamics for the endograph metric I: Equivalences with other metrics

T0 review · 1 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The endograph metric on fuzzy sets yields exactly the same topological dynamics as the stronger supremum, Skorokhod, and sendograph metrics, for transitivity, recurrence, Devaney chaos, and the specification property.

desk verdict Solid central equivalences for the endograph metric via a neat lemma; the new pointwise results in §3.3 rest on an unproved no-isolated-points fact deferred to the author's forthcoming paper. read the letter →

arxiv 2510.17990 v2 pith:OW2ZGDLF submitted 2025-10-20 math.DS

Antoni López-Martínez This is my paper · ORCID

classification math.DS MSC 37B0237B2047A1654A4054B20

keywords topologicaldynamicsfuzzydynamicalsystemsendographmetricZadehextensionFurstenbergfamiliestransitivityrecurrencespecificationproperty

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 establishes that the endograph metric, the coarsest of four standard metrics on the space of normal fuzzy sets, yields exactly the same topological dynamics as the stronger supremum, Skorokhod, and sendograph metrics for a precise list of properties: topological A-transitivity, (ℓ,A)-recurrence (covering ordinary recurrence and multiple recurrence), Devaney chaos, and the specification property. For each property, the induced Zadeh-extension system on fuzzy sets has the property exactly when the induced map on the compact hyperspace does. The entire argument rests on one lemma: a fuzzy set at endograph distance δ<1/2 from the characteristic function of a compact set K has all its α-level sets within Hausdorff distance δ of K, for α in an interval near 1. This answers an open question—whether endograph transitivity forces the original system to be weakly mixing—in the affirmative, and a block-family variation produces new equivalences for point-A-transitivity on separable complete metric spaces.

What carries the argument

The key machinery is Lemma 2.4, a quantitative bridge between the endograph metric and the Hausdorff metric: if a normal fuzzy set u satisfies d_E(χ_K,u)=δ<1/2 for a compact set K, then every α-level set u_α lies within Hausdorff distance δ of K for all α∈(δ,1−δ]. The lemma makes the endograph topology fine enough to carry return-set arguments: any orbit segment in a d_E-neighborhood of the characteristic function of a compact set projects down to an orbit segment in a Hausdorff neighborhood of the compact set, with the same return times, because the dynamics on α-levels is just the original map. The reverse inclusions τ_E⊂τ_S⊂τ_0⊂τ_∞ then give the opposite direction at the level of open set

What would settle it

Produce a counterexample to Lemma 2.4: a compact set K and a normal fuzzy set u with d_E(χ_K,u)=δ<1/2 but d_H(K,u_α)>δ for some α∈(δ,1−δ]. Since the lemma is used in every (vii)⇒(iii) direction, its failure would break the equivalences for transitivity, recurrence, Devaney chaos, and specification simultaneously. Alternatively, for the pointwise theorems, exhibit a separable complete metric space X where F_E(X) has an isolated point despite X having none (or vice versa), contradicting the deferred assumption.

Watch

Extended reading notes

Core claim

The core discovery is Theorem 3.1 with Theorems 3.3, 4.2, and 4.3: for every Furstenberg family A, the Zadeh-extension system (F_E(X), fhat) is topologically A-transitive if and only if the compact-hyperspace system (K(X), f) is topologically A-transitive, and the same holds for (ℓ,A)-recurrence, Devaney chaos, and the specification property. The engine is Lemma 2.4: if d_E(χ_K,u)=δ<1/2, then d_H(K,u_α)≤δ for every α∈(δ,1−δ]. Because α-levels commute with the dynamics, [fhat^n(u)]_α=f^n(u_α), this transfers return-set conditions from fuzzy systems to set-valued systems, completing the equivalence circle that was previously known only for the stronger metrics. A separate block-family argument

Load-bearing premise

The pointwise equivalences rest on a deferred claim (F_E(X) is either a singleton or has no isolated points for every metric space X), and the noncompact specification extension is asserted without proof; if either gap is not filled, those results collapse.

Editorial extensions

If this is right

Answered open question: endograph transitivity is equivalent to transitivity of the compact hyperspace system and to weak mixing of the original system; the answer to Question 1.1 and 1.2 is yes.
Recurrence unification: for any ℓ and Furstenberg family A, (ℓ,A)-recurrence of the endograph system is equivalent to that of the hyperspace system and of all finite products of the original system.
Devaney chaos: dense periodic points and transitivity hold for the endograph system if and only if they hold for the hyperspace system; in the linear Fréchet-space setting this also matches the original operator.
Specification property: holds for the endograph system if and only if it holds for the hyperspace system, and specification of the original system implies specification of all extended systems even when X is not compact.
Pointwise results: on separable complete metric spaces, for block Furstenberg families such as positive upper Banach density sets, point-A-transitivity is equivalent across the original, hyperspace, and all four fuzzy systems.

Reading between the lines

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

Since the endograph metric is strictly coarser than the others, these equivalences suggest that the listed properties are determined only by the coarse return-set structure of the system; a natural testable extension is whether the same Lemma 2.4 handles other open-return-set-definable properties such as mixing or weak mixing with prescribed return sets.
The block-family pointwise theorem implies that on Polish spaces a weakly mixing point-A-transitive map has a fuzzy point whose fhat-orbit hits every endograph-open set along a set in A; this offers a constructive way to build fuzzy sets with prescribed recurrence from compact-set data, which is not in the paper.
Lemma 2.4 is sharp at δ=1/2 in the sense that the chosen α-interval (δ,1−δ] becomes empty; it would be interesting to check whether the inequality still holds with δ=1/2 at α=1/2, which would delineate the exact threshold of the method.

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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

Quick take: this is a real paper. It closes the long-open gap for the endograph metric in the fuzzy-dynamics equivalence circle, and the engine is an elegant geometric lemma (Lemma 2.4) that transfers return-set arguments from endograph neighborhoods of characteristic functions to Hausdorff neighborhoods of compact sets. For A-transitivity, (ℓ,A)-recurrence, Devaney chaos, and specification, Theorems 3.1, 3.3, 4.2, and 4.3 are correctly argued and give the endograph metric the same status as d∞, d0, and dS. That is a clean, citable contribution.

The honest weakness is in §3.3. The point-transitivity and point-A-transitivity results — Corollary 3.5(b) and Theorem 3.7 — depend on the assertion that F_E(X) is either a singleton or has no isolated points for every metric space X. That fact is stated without proof and deferred to a self-cited forthcoming paper [39]. This is not cosmetic: the direction from point-transitivity to topological transitivity is used in those results, and it is false for arbitrary metric spaces with isolated points (e.g., the bi-infinite shift). The assertion is likely true — a second point in X lets you add a thin membership layer to a characteristic function and move it by an arbitrarily small endograph distance — but the manuscript does not include that argument. A referee cannot verify the new pointwise claims from the submitted text alone. The main equivalences do not rely on this fact, so the core of the paper survives.

There is a second, smaller gap in Corollary 4.4(a): the adaptation of Bauer–Sigmund to noncompact spaces is asserted to be 'easily adapted' without details. Since this concerns only the specification property of (K(X), f) and does not affect Theorem 4.3, I read the omission as minor but worth a referee note.

One more observation: the proof of Theorem 3.1 compresses the reduction to full families. I followed it, but it is the densest part of the paper and would benefit from an expanded write-up.

The audience is researchers in fuzzy and hyperspace dynamics; the main transfer lemma is reusable beyond this paper. Overall, the central equivalence results are correct and new, the key lemma is reusable, and the author is transparent about what is deferred. The paper deserves peer review, but the referee should ask for the no-isolated-points proof to be included (or for the pointwise claims to be explicitly conditional on it). If [39] appears with that proof, the pointwise section will stand. I would cite this for the main theorems and the lemma, and I might bring it to a reading group as a good example of a load-bearing structural lemma — with the gap as a talking point.

Recommendation: send it to a serious referee, with a clear request to verify or import the no-isolated-points proof before acceptance.

Referee Report

1 major / 4 minor

Summary. The paper studies a family of fuzzy dynamical systems (F(X), \hat f) on the space of normal fuzzy sets, comparing the endograph metric d_E with the supremum, Skorokhod, and sendograph metrics. The main technical contribution is Lemma 2.4, which states that if d_E(\chi_K, u) = \delta < 1/2, then d_H(K, u_\alpha) \le \delta for every \alpha \in (\delta, 1-\delta]. This lemma is used to transfer return-set arguments from d_E-neighbourhoods of characteristic functions to Hausdorff-neighbourhoods of compact sets. With it, the paper proves equivalences of topological \mathcal{A}-transitivity (Theorem 3.1), topological (\ell,\mathcal{A})-recurrence (Theorem 3.3), Devaney chaos (Theorem 4.2), and the specification property (Theorem 4.3) between (K(X), f), (F_\infty(X), \hat f), (F_0(X), \hat f), (F_S(X), \hat f), and (F_E(X), \hat f). It also formulates new point-\mathcal{A}-transitivity results (Theorem 3.7 and Corollary 3.9) for separable complete metric spaces and block Furstenberg families.

Significance. If the main equivalences hold, they unify and extend a substantial body of prior work by Jardón–Sánchez–Sanchis, Martínez-Giménez–Peris–Rodenas, Bartoll et al., and Álvarez et al., and they answer an open question (Question 1.1) about the endograph metric. The paper's core idea — Lemma 2.4 — is elegant, simple, and proved cleanly; it turns endograph closeness into Hausdorff closeness of level sets, which is exactly what is needed for return-set arguments. The reductions via the topology inclusions \tau_E \subset \tau_S \subset \tau_0 \subset \tau_\infty are sound, and the central equivalence theorems are well structured. The point-\mathcal{A}-transitivity results are the advertised new outcomes; however, as discussed in the major comment, they rely on an unproved and, in fact, false assertion about isolated points of F_E(X). The paper's central equivalences survive that issue, but the novelty advertised in the abstract is not fully supported as written.

major comments (1)

[§3.3] The text states that F_E(X) is either a singleton or has no isolated points, 'regardless of whether (X,d) itself has isolated points', and defers the proof to the forthcoming paper [39]. This assertion is false. For example, take X = {0,1} with the discrete metric and consider u = \chi_{\{0\}}. For every v \ne u in F(X), the endograph end(v) contains a point (y,\beta) with y \ne 0, and the d_E-distance from that point to end(\chi_{\{0\}}) is at least d(0,1) = 1. Hence d_E(u,v) \ge 1, so u is d_E-isolated. This property is load-bearing: it is used in Corollary 3.5(b)(vi)⇒(i) and in Theorem 3.7 (iii),(vii)⇒(i) to infer topological transitivity of (F_E(X), \hat f) from point-transitivity. Since the stated fact is false, those implications are not justified by the submitted manuscript. The author should provide a correct proof under the hypotheses of the theorems (e.g., when X is weakly-mixi

minor comments (4)

[Theorem 3.1] In the sentence 'given any arbitrary but fixed n∈A one can find compact sets...', the symbol 'A' should be 'B'; the argument is intended for an element of the intersection B.
[§2.1] The definition contains the phrase 'but A ≠ P(N0)'. This is confusing: the earlier condition ∅∉A already excludes A=P(N0). The phrase seems redundant and should be removed or clarified.
[§4.2] The claim that the Bauer–Sigmund compactness argument 'can be easily adapted to general metric spaces' is stated without proof. Since uniform continuity is not available in the noncompact setting, this is not a completely routine adaptation. Please include a proof or a precise reference.
[Lemma 3.6] The proofs of parts (d) and (e) omit several details ('we omit the routine verification', 'arguing as in [2, Theorem 4.1]'). These steps are important for constructing an \mathcal{A}-recurrent compact set / fuzzy set from an \mathcal{A}-recurrent point in a product. Please expand them so that the construction is verifiable from the submitted text.

Circularity Check

1 steps flagged · score 4.0 of 10

Pointwise endograph results rest on an unproved no-isolated-points assertion deferred to the author's own [39].

self citation load bearing [Section 3.3, paragraph before Corollary 3.5; used in the proof of Corollary 3.5(b) and Theorem 3.7]
"Moreover, one can show with not too much difficulty that for any metric space (X, d), the associated fuzzy metric space FE(X) is either a singleton (precisely when X is a singleton) or has no isolated points, regardless of whether (X, d) itself has isolated points (a proof of this fact will appear in [39])."

This is the only support for the implication 'FE(X) point-transitive ⇒ FE(X) topologically transitive' used explicitly in Corollary 3.5(b)(vi)⇒(i) and, through Corollary 3.5(b), in Theorem 3.7. The proof is not given; it is deferred to the author's own forthcoming paper [39]. Thus the new point-transitivity and point-A-transitivity conclusions are carried by a load-bearing self-citation rather than by a proof contained in this manuscript.

full rationale

No definitional circularity, fitted-input-called-prediction, renaming, or ansatz-smuggling is present. The main equivalences (Theorems 3.1, 3.3, 4.2 and 4.3) are genuine deductions: the key Lemma 2.4 is a new geometric argument converting endograph-neighborhoods of characteristic functions into Hausdorff-neighborhoods of their slices, and the return-set inclusions in (vii)⇒(iii), (vi)⇒(ii), and (v)⇒(i) do not reduce to their hypotheses. Reliance on [2,4,29,41] for the d∞/d0/sendograph parts is citation of independent published work. The only circularity concern is the asserted no-isolated-points structure of FE(X), which is load-bearing for the pointwise results in Section 3.3 but is explicitly deferred to the author's companion paper [39]; under the reviewing rules this is a load-bearing unverified self-citation. Corollary 4.4(a)'s 'proof can be easily adapted' to noncompact spaces is a completeness gap, not circularity. Hence the central claims retain independent content, but the pointwise new results are not fully self-contained.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation is a geometric transfer lemma with no fitted constants and no new entities. The load-bearing assumptions are the standard compactness and level-set facts about normal fuzzy sets, the well-known continuity of the Zadeh extension, and two technical facts: the no-isolated-points property of F_E(X) and the noncompact adaptation of the Bauer–Sigmund specification argument.

assumptions (5)

domain assumption For each ρ ∈ {d∞, d0, dS, dE}, the Zadeh extension hat f: (F(X), ρ) → (F(X), ρ) is continuous.
Invoked throughout to regard (F(X), hat f) as a dynamical system; quoted as well-known from [28,33] in Section 2.2.
standard math For every u ∈ F(X) and α > 0, the α-level u_α is a non-empty compact set and d_H is a metric on K(X).
Used in Lemma 2.4 and in every transfer from fuzzy orbits to compact-set orbits; follows from the definition of normal fuzzy sets but is load-bearing.
standard math The identity d_E(u,v) ≤ d_S(u,v) ≤ d_0(u,v) ≤ d_∞(u,v), hence τ_E ⊂ τ_S ⊂ τ_0 ⊂ τ_∞.
Used repeatedly in the trivial directions (iv)⇒(v)⇒(vi)⇒(vii); stated as well-known in Section 2.2.
domain assumption F_E(X) is either a singleton or has no isolated points.
Stated in Section 3.3 with proof deferred to the author's forthcoming [39]; used in Corollary 3.5(b) and Theorem 3.7 to pass from point-transitivity to topological transitivity.
domain assumption The Bauer–Sigmund proof that specification on (X,f) implies specification on (K(X),f) adapts from compact to arbitrary metric spaces.
Assumed in Corollary 4.4(a) with the comment that the proof 'can be easily adapted' by using continuity instead of uniform continuity; no adaptation is shown.

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Pith. "Pith review of Topological dynamics for the endograph metric I: Equivalences with other metrics." pith.science (2026). https://pith.science/paper/OW2ZGDLF

@misc{pith2026251017990,
  author       = {Pith},
  title        = {Pith review of: Topological dynamics for the endograph metric I: Equivalences with other metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OW2ZGDLF}},
  note         = {Machine review of arXiv:2510.17990}
}

abstract

Given a dynamical system $(X,f)$ we investigate several topological dynamical properties for its Zadeh extension $(\mathcal{F}(X),\hat{f})$ endowed with the endograph metric $d_{E}$. In particular, we prove that for topological $\mathcal{A}$-transitivity, topological $(\ell,\mathcal{A})$-recurrence, Devaney chaos, and for the specification property, the endograph metric behaves similarly to the supremum metric $d_{\infty}$, the Skorokhod metric $d_{0}$ and the sendograph metric $d_{S}$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-$\mathcal{A}$-transitivity.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Li-Yorke chaos on fuzzy dynamical systems
math.DS 2025-10 conditional novelty 6.0 of 10

Li-Yorke and distributional chaos transfer from a dynamical system to its hyperspace and fuzzy-set extensions, while a new Cantor-dense variant transfers back under completeness and linearity assumptions.
Topological dynamics for the endograph metric II: Extremely radical properties
math.DS 2025-10 accept novelty 6.0 of 10

For the endograph-metric Zadeh extension, contractivity iff f is constant, expansivity-type iff X is a singleton, and chain recurrence/transitivity/mixing iff f has dense range.

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [39]

L´ opez-Mart ´ ınez

A. L´ opez-Mart ´ ınez. Topological dynamics for the endograph metric II: Extremely radical properties.arXiv preprint, to appear

[2] [1]

Abakumov and A

E. Abakumov and A. Abbar. On several dynamical properties of shifts acting on directed trees.Canad. J. Math., (2025), 1–30

2025

[3] [2]

´Alvarez, A

I. ´Alvarez, A. L´ opez-Mart ´ ınez, and A. Peris. Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems.Fuzzy Sets Syst.,506(109296) (2025), 1–13

2025

[4] [3]

J. Banks. Chaos for induced hyperspace maps.Chaos Solitons Fractals,25(2005), 681–685

2005

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Bartoll, F

S. Bartoll, F. Mart ´ ınez-Gim´ enez, A. Peris, and F. Rodenas. Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems.Axioms,11(12) (2022), 733

2022

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