Topological dynamics for the endograph metric II: Extremely radical properties

Antoni L\'opez-Mart\'inez

REVIEW 2 major objections 5 minor 1 cited by

Topological dynamics for the endograph metric II: Extremely radical properties

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The endograph metric makes fuzzified dynamics contractive only for constant maps, expansive only on one-point spaces, and chain-mixing exactly when the original map has dense range.

desk verdict A solid, no-nonsense contribution to the fuzzy-hyperspace dynamics niche; the main dichotomies hold up, and the imported lemma is fine. read the letter →

arxiv 2510.19337 v2 pith:FYJNDV4W submitted 2025-10-22 math.DS

Antoni López-Martínez This is my paper · ORCID

classification math.DS MSC 37B0237B0537B6554A4054B20

keywords topologicaldynamicsfuzzydynamicalsystemsendographmetricZadehextensionexpansivepropertieschainrecurrenceshadowingpropertynormalsets

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 asks how the standard fuzzification of a continuous map behaves when the space of fuzzy sets is equipped with the endograph metric, which measures the Hausdorff distance between the regions under the graphs of two fuzzy sets. The author establishes that this metric makes the dynamical behaviour extreme: the fuzzy system is contractive exactly when the original map is constant, and it is expansive, expanding, or positively expansive exactly when the space is a singleton. More strikingly, chain recurrence, chain transitivity, and chain mixing all become equivalent to a single condition on the original map — that it has dense range. The paper also corrects a published shadowing result by giving counterexamples and proving that, for a dense-range map, the endograph fuzzy system cannot have finite shadowing unless the original system is topologically mixing.

What carries the argument

The key object is the endograph metric d_E on the space F(X) of normal fuzzy sets: end(u) = {(x,α)∈X×I: u(x)≥α} is the closed region under a fuzzy set's graph, and d_E(u,v) is the Hausdorff distance between end(u) and end(v) in X×[0,1]. Its strange topology — no isolated points, path-connectedness — comes from the fact that every endograph contains X×{0}. The engine of the proofs is Lemma 2.5, which converts a small d_E-distance between a crisp set χ_K and a fuzzy set u into small Hausdorff distance between K and the α-level u_α for every intermediate level; this bridge lets the author transfer dense range and shadowing between the fuzzy system and the classical hyperspace.

What would settle it

Compute d_E and d_H for X={0,1} with the discrete metric, K={0}, and u with u(0)=1, u(1)=β for β∈(0,1/2). Lemma 2.5 predicts that d_H({0},u_α)=0 for α>β and that the exceptional level α≤β lies outside the interval (δ,1-δ] where δ=d_E(χ_K,u)=β. Verifying this boundary directly is the decisive check: if any α∈(β,1-β] had d_H({0},u_α)>0, the lemma — and with it the dense-range characterizations of Theorem 4.3 — would fail.

Watch

Extended reading notes

Core claim

The paper's central discovery is a series of 'if and only if' collapses. For the Zadeh extension fhat acting on normal fuzzy sets with the endograph metric d_E, the system (F_E(X), fhat) is chain recurrent, chain transitive, and chain mixing if and only if the original continuous map f has dense range. In the same setting, contractivity of the fuzzy system occurs exactly when f is constant, and expansiveness, expanding, or positive expansiveness occurs exactly when X is a singleton. The shadowing property sits in between: if f has dense range and (X,f) is not topologically mixing, then (F_E(X), fhat) lacks even finite shadowing; if f is contractive and some iterate f^k(X) is bounded, then (F

Load-bearing premise

A single imported metric lemma (Lemma 2.5) carries the weight: it asserts that a fuzzy set that is d_E-close to a crisp set must have all intermediate levels Hausdorff-close to the crisp set; if that fails, the chain-recurrence and shadowing collapses break.

Editorial extensions

If this is right

If f has dense range, then (F_E(X), fhat) is chain mixing, hence chain transitive and chain recurrent; conversely, any chain recurrent fuzzy system forces f(X) dense.
For any nonconstant f, none of the fuzzy systems with the Skorokhod, sendograph, or endograph metrics can be contractive; for any nonsingleton X, none of them can be expansive, expanding, or positively expansive.
The metric space F_E(X) has no isolated points whenever X is not a singleton (and is path-connected), so isolated-point arguments cannot be used in the endograph setting.
A published equivalence claiming that finite shadowing passes between (X,f), (K(X),f), and (F0(X),fhat) is only partially true; the correct equivalence for finite shadowing involves F∞, and d0, dS, dE require extra hypotheses such as contractivity with bounded eventual image.
For dense-range non-mixing systems, (F_E(X), fhat) cannot have even finite shadowing, so many natural surjective or dense-range systems fail a common orbit-tracing property.

Reading between the lines

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

The pattern suggests d_E is dynamically 'forgetful': long-term orbit-tracing properties are determined almost entirely by the image of f rather than by the fine structure of X, so analogous reductions might hold for other chain-type or tracing notions not treated here.
Because d0 and dS also appear in the counterexamples and in the shadowing results, the phenomenon is not unique to the endograph metric; a promising test would be to see whether the Skorokhod metric admits similar dense-range collapses for chain notions.
The imported Lemma 2.5 is the single point to stress-test; if one replaces d_E by a truncated or level-limited variant, the equivalences may fail, giving a way to build metrics with intermediate behaviour.
For maps lacking dense range, one could try to characterize chain recurrence of F_E via the closure of f(X), perhaps recovering a graded version of the dense-range collapse.

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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

Dear colleague,

You can take this paper at face value: it is a substantive, honest continuation of the author's Part I, and the 'extremely radical' claims in the title are real. The paper proves that for the endograph metric on normal fuzzy sets, contractivity of the Zadeh extension forces the base map to be constant, and any expansive/expanding/positively expansive behavior forces the base space to be a singleton. It also gives a clean characterization of chain recurrence/transitivity/mixing on F_E(X) as equivalent to the base map having dense range, and it fixes the flawed shadowing theorem from Bartoll et al. by counterexample plus a corrected result. Those are genuinely new results and solve several open problems from the literature. The proofs are explicit and, as far as I can tell, correct.

The main thing that looked fragile was Lemma 2.5, imported from the companion paper, which converts d_E-density into Hausdorff-dimensional-level approximation. I checked it directly: the lemma is true, and the uses in Lemma 4.4 and Theorem 5.3 are legitimate. So the central equivalences hold together. What remains is a set of minor issues: a fair number of typographical slips (e.g., 'ca be used' in Remark 3.2, and a concatenation in Theorem 4.1 that is hard to parse), and the proof of Theorem 5.6 is quite long and case-heavy. The paper would benefit from a round of careful editing, but the arguments are recoverable.

One thing to keep in mind: the paper leans on the author's own Part I for key background and for Lemma 2.5. That is not circular, but it means the referee should look at both parts together. The citation pattern to the rest of the literature is appropriate. The author is not overclaiming; the conclusions are stated with the right scope.

This is a niche paper — it will matter to people working on fuzzy/hyperspace dynamics, not to a broad dynamical systems audience. But within that niche it's important: it resolves open problems and fixes a published theorem. I'd send it to a serious referee, and I'd probably cite it if I worked in the area.

Recommendation: engage with it; the referee should be someone who can check the long shadowing proof and who has access to [27].

Referee Report

2 major / 5 minor

Summary. The paper studies the Zadeh extension of a continuous self-map f on a metric space (X,d) to the space F(X) of normal fuzzy sets, endowed with the supremum, Skorokhod, sendograph, and endograph metrics. Its main claims are: (a) the systems (F_0(X), fhat), (F_S(X), fhat), (F_E(X), fhat) are contractive iff f is constant, and expansive/expanding/positively expansive iff X is a singleton (Theorem 3.1); (b) chain recurrence, chain transitivity, and chain mixing for (F_E(X),fhat) are all equivalent to f having dense range (Theorem 4.3); and (c) several shadowing results, including counterexamples to a claim in [5] and a positive shadowing theorem for contractions with a bounded iterate (Theorems 5.3, 5.4, 5.6). The paper also records an isolated-points dichotomy for the endograph metric (Lemma 2.6). The exposition is careful and the central constructions, especially in Lemma 3.3 and Theorem 4.3, are explicit and mostly rigorous.

Significance. If correct, the results give a complete and rather striking picture of the dynamical behaviour of fuzzy extensions with the endograph metric, resolving open problems from [18] and [20] and correcting [5, Theorem 5]. The paper's explicit chain construction in Theorem 4.3 and the distance-normalization technique in Lemma 3.3 are valuable and verify the claimed dichotomies without hidden parameters. The treatment of non-compact metric spaces is a genuine extension of earlier work. The main theorems, if sound, are significant for the fuzzy-dynamics community. However, two proof issues — one in the standalone Lemma 2.6 and one in the proof of Theorem 5.6 — need attention before the paper is fully convincing.

major comments (2)

[Lemma 2.6, proof, Case 2] The proposed perturbation w := u · χ_{X\B_d(x,δ)} is not guaranteed to lie in B_E(u,ε). For X={0,1} with the discrete metric, u=χ_X and x=0, every δ<1 gives w=χ_{1}; by Proposition 2.4(d), d_E(χ_X,χ_{1})=1, so w∉B_E(u,ε) for 0<ε<1. Thus the proof of part (c) as written fails. The same kind of support-deletion construction in part (b), Case 2, also requires a careful choice of δ and is not valid for arbitrary δ<ε. Since this lemma is not used later, the central theorems are unaffected, but the proof should be replaced — for instance by lowering membership values on a small ball rather than deleting the support.
[Theorem 5.6, proof, second part] The proof covers complete spaces and then 'bounded but not complete' spaces. However, the theorem's hypothesis — f^k(X) bounded for some k and f contractive — does not imply that X is bounded. For example, X=Q with f(x)=c x/(1+|x|), 0<c<1, is a contraction with f(X) bounded and X unbounded and incomplete. The completion argument can be extended to this case because f*^k(X*) is contained in the closure of f^k(X) and hence bounded, but the manuscript does not state or prove this. As written, Theorem 5.6 is not proved for all metric spaces covered by its statement. Please add the missing case or restrict the theorem accordingly.

minor comments (5)

[Section 5.2, definition of topologically mixing] The definition reads 'f^n(U) ∪ V ≠ ∅', which is trivially true; it should be 'f^n(U) ∩ V ≠ ∅'.
[Theorem 5.6, Case 2 of the proof] The intermediate displayed inclusion contains indexing slips: the term with l=0 uses a negative radius '(l−1)δ_ε', and the inclusions following the choice of x_{j+1}, y_{j+1} refer to end(fhat^j(u)) where the conclusion concerns end(fhat^{j+1}(u)). These are typos, but should be corrected for readability.
[Theorem 4.1, proof (i)⇒(ii)] In the induction step for chain weak-mixing, the concatenated sequences are correct, but the second displayed sequence would be clearer if it were explicitly described as an (N+1)-tuple with the first coordinate repeated; the current typesetting invites misreading.
[References] References [2] and [23] are both assigned the same arXiv number (2411.17037v1); one of them appears to be a typo.
[Remark 3.2] Typo: 'ca be used' should be 'can be used'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central theorems are proved by explicit constructions and independent metric lemmas.

full rationale

The main results (Theorem 3.1, Theorem 4.3, Theorems 5.3–5.6) are derived by direct, non-tautological arguments rather than by fitting, normalization, or definitional identification. Theorem 3.1 is proved through Lemma 3.3, which explicitly constructs two fuzzy sets u and u_k whose d_0, d_S, and d_E distances are 1/k while the distances of their images under the Zadeh extension are also controlled; this yields a concrete contradiction with contractivity or expansivity. Theorem 4.3 gives an explicit d_E-δ-chain of any prescribed length between arbitrary fuzzy sets u and v when f has dense range, and the converse relies on the standard fact that a chain recurrent map must have dense range together with Lemma 4.4. The only externally imported ingredient is Lemma 2.5, cited from the author's companion paper [27]; it is an elementary, parameter-free metric fact about the endograph metric and is independent of the chain, shadowing, or expansivity conclusions it supports. Likewise, the use of [27, Theorem 3.1] in Theorem 5.4 is a prior external equivalence used inside a new finite-shadowing argument, not a re-use of the theorem being proved. There are no fitted parameters, no predicted quantities that are equal by construction to quantities already used as inputs, and no uniqueness theorem or ansatz imported from the author's own work as a substitute for proof. The derivation chain is therefore not circular.

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

No free parameters fitted to data and no invented entities. The mathematics relies on the standard fuzzy-set framework, known hyperspace facts, and one key lemma from the author's companion paper [27].

assumptions (5)

domain assumption Normal fuzzy sets: F(X) consists of upper-semicontinuous u:X→[0,1] with compact u_0 and nonempty u_1; d_E is the Hausdorff distance between endographs in X×[0,1].
The entire paper lives in this framework, inherited from [20,21,30].
domain assumption Zadeh extension fhat exists and is continuous for d∞, d0, dS, dE, and [fhat(u)]_α = f(u_α).
Used throughout; cited to [20, Propositions 3.1 and 4.9] and [26].
domain assumption Lemma 2.5 ([27, Lemma 2.4]): if δ=d_E(χ_K,u)<1/2 then d_H(K,u_α)≤δ for all α∈]δ,1−δ].
Imported from the author's companion paper; load-bearing for Lemma 4.4(vi)⇒(ii) and Theorem 5.3(c).
standard math Standard hyperspace facts: finite subsets are d_H-dense in K(X); Hausdorff metric inequalities; Vietoris topology basics.
Used in Theorem 4.1 and elsewhere; cited to [17].
standard math Known shadowing equivalences and lemmas: [12, Theorem 3.4] arguments for finite shadowing on hyperspaces, [31] linear shadowing, contraction shadowing folklore, and [27, Theorem 3.1] for topological mixing.
Applied in Section 5 to pass shadowing between (X,f), (K(X),f) and the fuzzy extensions.

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Pith. "Pith review of Topological dynamics for the endograph metric II: Extremely radical properties." pith.science (2026). https://pith.science/paper/FYJNDV4W

@misc{pith2026251019337,
  author       = {Pith},
  title        = {Pith review of: Topological dynamics for the endograph metric II: Extremely radical properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FYJNDV4W}},
  note         = {Machine review of arXiv:2510.19337}
}

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 some contractive and expansive properties, for chain recurrence, chain transitivity and chain mixing, and for the shadowing property, the endograph metric behaves in an extremely radical way. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes concerning the chain-type notions considered and the shadowing property.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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
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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.

Reference graph

Works this paper leans on

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Pith tools

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

[1] [27]

L´ opez-Mart ´ ınez

A. L´ opez-Mart ´ ınez. Topological dynamics for the endograph metric I: Equivalences with other metrics.arXiv preprint, arXiv:2510.17990, 23 pages

arXiv

[2] [5]

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

[3] [18]

Jard´ on and I

D. Jard´ on and I. S´ anchez. Expansive properties of induced dynamical systems.Fuzzy Sets Syst.,425(2021), 48–61

2021

[4] [20]

Jard´ on, I

D. Jard´ on, I. S´ anchez, and M. Sanchis. Some questions about Zadeh’s extension on metric spaces.Fuzzy Sets Syst., 379(2020), 115–124

2020

[5] [1]

´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. 30

2025

[6] [3]

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

2005

[7] [4]

M. F. Barnsley.Fractals everywhere. Second edition. Morgan Kaufmann, Elsevier, 1988

1988

[8] [6]

A. D. Barwell, C. Good, P. Oprocha, and B. E. Raines. Characterizations ofω-limit sets in topologically hyperbolic systems.Discret. Contin. Dyn. Syst.,33(2013), 1819–1833

2013