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REVIEW 3 major objections 4 minor 41 references

Li-Yorke chaos on fuzzy dynamical systems

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Li-Yorke chaos passes to fuzzy extensions of a dynamical system, but only a strengthened variant passes back.

desk verdict Solid systematic extension of [34] with good estimates and counterexamples, but the advertised reverse-transfer rests on an unproved external equivalence [26, Prop 3.5] that the authors should pin down. read the letter →

arxiv 2510.20240 v2 pith:S4LAGWBY submitted 2025-10-23 math.DS

Illych Álvarez , Antoni López-Martínez This is my paper · ORCID
classification math.DS MSC 37B0237B0537B2554A4054B20
keywords Li-YorkechaosdistributionalfuzzydynamicalsystemsZadehextensionhyperspacesCantor-denseproximalitysensitivity
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 what happens to Li-Yorke chaos when a dynamical system is extended to its space of compact subsets and then to its space of normal fuzzy sets. It proves that all main variants—Li-Yorke, mean Li-Yorke, and distributional chaos types DC1 through DC3, plus uniform versions—transfer forward along these extensions for the supremum, Skorokhod, and sendograph metrics, and largely for the endograph metric. The converse transfer fails in general, and the paper supplies counterexamples where the fuzzy system is chaotic though the original system has almost no scrambled pairs. To repair the converse, it introduces Cantor-dense Li-Yorke chaos, which does transfer back from the fuzzy system to the hyperspace when the underlying space is complete. For linear operators on Fréchet spaces, several of these properties become equivalent to non-equicontinuity.

What carries the argument

The argument runs on two embeddings: x ↦ {x} (points to singletons) and K ↦ χ_K (compacta to their characteristic fuzzy sets). These are isometries from (X,d) to (K(X),d_H) and from (K(X),d_H) into the fuzzy spaces with supremum, Skorokhod, and sendograph metrics, so scrambled pairs lift unchanged. For the endograph metric, which only sees distances capped at 1, the paper develops level-set estimates (Lemmas 2.6 and 2.7) that translate sendograph and endograph distance into Hausdorff distance between α-levels. The strengthened converse uses a recent characterization: on completely metrizable spaces, Cantor-dense Li-Yorke chaos is equivalent to having a dense proximal relation together with s

What would settle it

Find a complete metric space (X,d) and a continuous f such that (F∞(X),f_hat) is Cantor-dense Li-Yorke chaotic but (K(X),f_bar) is not; this would disprove the main reverse-transfer theorem. Alternatively, exhibit a completely metrizable system that is both sensitive and has a dense proximal relation but fails to have scrambled Cantor sets in every open set—this would invalidate the equivalence on which the proof leans.

Watch

Extended reading notes

Core claim

The central claim is that chaos survives the trip to larger spaces in one direction but not the other. Given a continuous map f on a metric space X, the paper shows that if (X,f) is Li-Yorke chaotic (in any of its standard variants, including mean Li-Yorke and distributional chaos DC1–DC3, or their uniform versions), then so is the induced system on the hyperspace of non-empty compact subsets with the Hausdorff metric, and so is the system on normal fuzzy sets equipped with the supremum, Skorokhod, or sendograph metric. The endograph metric is more delicate and needs boundedness assumptions. Conversely, chaos in the fuzzy system does not in general force chaos in the hyperspace or in X; expl

Load-bearing premise

The reverse-transfer theorem rests on a cited equivalence: on completely metrizable spaces, Cantor-dense Li-Yorke chaos is equivalent to having a dense proximal relation together with sensitivity; if that equivalence fails—or if (F∞(X),d∞) and (F0(X),d0) are not complete—the strengthened converse does not follow.

Editorial extensions

If this is right

  • If (X,f) exhibits any of LYC, MLYC, DC1, DC1½, DC2, DC2½, DC3, or their uniform versions, then so do (K(X),f_bar), (F∞(X),f_hat), (F0(X),f_hat), and (FS(X),f_hat).
  • The converse fails: there are systems whose fuzzy extension is U-DC1 and U-MLYC while every LY- or D3-scrambled subset of X has at most two points, and every such subset of K(X) is finite.
  • For the endograph metric, transfer holds for LY, DC1, DC1½, DC2, DC2½ and their uniform versions, but MLYC and DC3 require boundedness or diameter≤1 assumptions, and the paper gives examples showing these assumptions cannot be dropped.
  • Cantor-dense Li-Yorke chaos transfers from (F∞, F0, FS) to (K(X)) when X is complete, giving a negative answer to the open question for this strengthened notion.
  • For linear operators on Fréchet spaces, under a dense set of vectors with a null sub-orbit, U-LYC and CD-LYC become equivalent to non-equicontinuity across all the relevant extensions.

Reading between the lines

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

  • The positive-transfer results suggest a monotonicity principle: Li-Yorke chaos is robust under passing to the 'cloud' of a system—points, subsets, fuzzy sets—while the inverse problem is where the real structure lives; the paper's CD-LYC is a natural candidate for a chaotic core that survives both directions.
  • The counterexamples rely on carefully crafted metrics where proximal pairs are rare; one could test whether the same separation between (X,f) and (FE(X),f_hat) occurs for simpler metrics on interval maps, where distributional chaos may be generic.
  • A testable extension: for non-normal fuzzy sets (dropping normality), the transfer results might fail at the first step, since the characteristic-function embedding relies on compact support and a non-empty 1-level; checking the paper's lemmas in that setting would delimit the boundary.
  • Because the converse transfer leans on an external equivalence, a proof that bypasses that equivalence might weaken the completeness assumptions or extend to non-metrizable uniform spaces.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies transfer of Li-Yorke-type chaos between a metric dynamical system (X,f), its hyperspace extension (K(X), f_bar) with the Hausdorff metric, and the fuzzy extension (F(X), f_hat) with the supremum, Skorokhod, sendograph, and endograph metrics. The main forward results (Theorems 3.1 and 3.2) show that LY, mean LY, and distributional chaos of types DC1–DC3 transfer from (X,f) to (K(X),f_bar) and from (K(X),f_bar) to (F(X),f_hat), with the endograph metric requiring boundedness or diameter conditions. Three counterexamples answer Question 1.1 positively and show optimality of boundedness assumptions. The paper then introduces Cantor-dense Li-Yorke chaos (CD-LYC) and, using proximality and sensitivity results from Section 4, proves reverse transfer from fuzzy systems to the hyperspace under completeness (Theorems 5.2 and 5.5), and equivalences for linear operators on Fréchet spaces.

Significance. If the results hold, the paper gives a systematic and fairly complete picture of how the main Li-Yorke-type chaotic properties behave under hyperextension and fuzzification. The detailed metric estimates (3.4) and (3.5) are a genuine technical contribution, and the counterexamples are structurally coherent and directly address a question left open in [34]. The proximality and sensitivity transfer results in Section 4 are also new and are proved with care, including a corrected proof of a point left unclear in [20]. The CD-LYC reverse-transfer results are potentially significant, but their current form depends on an external characterization from [26] whose exact scope is not verified in the manuscript.

major comments (3)
  1. [§5.1, Theorem 5.2; §5.2, Theorem 5.5(b)] The reverse-transfer claims for CD-LYC rest entirely on [26, Proposition 3.5], quoted as valid for arbitrary continuous maps on completely metrizable spaces. However, the title of [26] indicates a focus on endomorphisms of completely metrizable groups and linear operators on Fréchet spaces, and the spaces K(X), F∞(X), F0(X) carry no group or linear structure in general. The authors should either reproduce the exact statement of [26, Proposition 3.5] and confirm that it applies to arbitrary continuous maps, or provide a direct proof of the dense proximal relation + sensitivity ⇒ CD-LYC implication. Without this, the conclusions of Theorem 5.2 and Theorem 5.5(b)(iii)–(v) do not follow from the material proved in the paper.
  2. [§3.2, Examples 3.4, 3.5, 3.6] In each example the authors write 'We leave to the reader checking that (X,d) is a metric space'. The metrics are piecewise-defined using sets A with prescribed density properties, and the triangle inequality is not immediate, particularly for triples of points sharing a first coordinate where one lies in A and another not. Since these examples are load-bearing for the claimed failures of converse transfer and for the sharpness of boundedness assumptions, the verification should be included or at least sketched. This is fixable but is currently a gap in the proof of the counterexamples.
  3. [§5.2, proof of Theorem 5.5(b)] The step '(i)⇒(iii),(iv),(v)' applies [26, Proposition 3.5] to (K(X),T), (F∞(X), f_hat), and (F0(X), f_hat) after establishing dense proximal relation and sensitivity. Besides the external-proposition concern raised above, the completeness of F∞(X) and F0(X) is cited from [25] and [15] without any statement of the results. The authors should state explicitly the reference results used, because the CD-LYC conclusion requires the full force of the Mycielski-based equivalence, not merely the existence of a scrambled Cantor set in one open set.
minor comments (4)
  1. [§2.2, definition of d_E] There is a typo: the endograph metric is defined as 'the Hausdorff distance d_H(end(u),end(u))', which should be d_H(end(u),end(v)).
  2. [§3.1, Theorem 3.1(c)] The use of ε′:=min{ε,1} is clear for D1 and D2, but the statement lists LY without a parameter; this creates a small notational mismatch. Consider reformulating the LY case separately.
  3. [§5.2, Theorem 5.5(b)] The set X0 is defined using 'some increasing sequence (n_k)', but the statement does not specify whether the same sequence is used for all points. This should be clarified, since the proof of density of the proximal relation uses X0^2 ⊂ Prox(f,d).
  4. [§3.2, Example 3.5] The computation of the liminf for the scrambled set S omits some justifications in the inequalities involving [n] = sqrt(log2 n); for example, the equality of sums over A ∩ [1,n−1] with powers of 2 is not fully spelled out. A short explanation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transfer theorems are derived from definitions and direct estimates; the reverse-transfer step uses an external characterization ([26, Prop. 3.5]) and the paper's self-citations are auxiliary, not load-bearing.

full rationale

The core results (Theorems 3.1 and 3.2) are not circular. They construct scrambled sets explicitly, either by isometric embeddings (ι1, ι2) or by the family u_α = max{χ_K, αχ_L}, and they verify the required metric estimates directly in Eqs. (3.4) and (3.5). The conclusions are not assumed in the hypotheses. The converse examples are independent constructions rather than restatements of the theory. The reverse-transfer results in Theorem 5.2 and parts of Theorem 5.5 rely on the externally cited characterization [26, Proposition 3.5] (dense proximality plus sensitivity is equivalent to Cantor-dense Li-Yorke chaos on completely metrizable spaces). That is an external theorem, not a self-citation and not an input of the present paper; any doubt about its validity or scope is a correctness/support risk, not a circularity. The paper's self-citations ([32, Lemma 2.6] and [33]) are used only as auxiliary technical support and are not the load-bearing premise of the main claims; Lemma 2.6 concerns a peripheral endograph-metric estimate and is even largely replaceable by the paper's own Lemma 2.7. There are no fitted parameters, no predictions that reduce to their fits, and no uniqueness argument imported from the authors' prior work. Omitted verifications, such as 'We leave to the reader checking that (X,d) is a metric space', are expositional gaps, not circular steps. Overall, the derivation chain is not circular.

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

The paper introduces no new physical or mathematical entities; it uses standard fuzzy-set and hyperspace constructions. The main external inputs are previously established level-set identities, one lemma from the same author's earlier work, the completeness of the fuzzy metric spaces, and the [26] characterization of CD-LYC. There are no fitted parameters.

assumptions (5)
  • domain assumption Zadeh extension f_hat maps F(X) to F(X) and satisfies the level-set identity [f_hat(u)]_alpha = f(u_alpha) for all alpha (Proposition 2.4).
    Imported from [21,22,35]; underpins essentially all computations in Sections 3 and 4.
  • domain assumption Lemma 2.6 from [32]: if d_E(chi_K, u) < 1/2, then d_H(K, u_alpha) is controlled on an interval of levels.
    Used in the proof of Theorem 4.2, Case 3; taken from a prior paper by the same second author.
  • domain assumption [26, Proposition 3.5]: on completely metrizable spaces, CD-LYC is equivalent to having a dense proximal relation together with sensitivity.
    Load-bearing for Theorem 5.2 and the CD-LYC parts of Theorem 5.5; quoted rather than proved.
  • domain assumption [5, Theorems 9 and 15]: with span(NS(T)) dense, a non-equicontinuous operator on a Fréchet space is Li-Yorke chaotic and admits an irregular vector.
    Used in the proof of Theorem 5.5(a) to pass from non-equicontinuity to U-LYC.
  • domain assumption The spaces (K(X), d_H), (F_infty(X), d_infty), and (F_0(X), d_0) are complete when X is complete.
    Quoted from [18,25,15]; required to apply [26, Proposition 3.5] in Section 5.

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Pith. "Pith review of Li-Yorke chaos on fuzzy dynamical systems." pith.science (2026). https://pith.science/paper/S4LAGWBY

@misc{pith2026251020240,
  author       = {Pith},
  title        = {Pith review of: Li-Yorke chaos on fuzzy dynamical systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4LAGWBY}},
  note         = {Machine review of arXiv:2510.20240}
}
abstract

Given a dynamical system $(X,f)$ we investigate how several variants of Li-Yorke chaos behave with respect to the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, where $\overline{f}$ is the hyperextension of $f$ acting on the space $\mathcal{K}(X)$ of non-empty compact subsets of $X$, and where $\hat{f}$ denotes the Zadeh extension of $f$ acting on the space $\mathcal{F}(X)$ of normal fuzzy subsets of $X$. We first prove that the main variants of Li-Yorke chaos transfer from $(X,f)$ to $(\mathcal{K}(X),\overline{f})$ and from $(\mathcal{K}(X),\overline{f})$ to $(\mathcal{F}(X),\hat{f})$, but that the converse implications do not hold in general. However, combining the notions of proximality and sensitivity we introduce Cantor-dense Li-Yorke chaos, and we prove that this strengthened variant of chaos does transfer from $(\mathcal{F}(X),\hat{f})$ to $(\mathcal{K}(X),\overline{f})$ under natural assumptions.

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