Li--Yorke Chaos Along Any Infinite Sequence: Relative Mixing, Sofic and Rokhlin Entropy

Chunlin Liu

REVIEW 3 major objections 4 minor 17 references

Li--Yorke Chaos Along Any Infinite Sequence: Relative Mixing, Sofic and Rokhlin Entropy

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Positive sofic entropy forces Li–Yorke scrambled Cantor sets along every prescribed infinite sequence of group elements.

desk verdict Genuinely answers Huang–Li–Ye's open question with a new and clean relative-mixing mechanism; the main proof is sound, and the only real caveat is the unverified applicability of two cited external theorems. read the letter →

arxiv 2607.20735 v1 pith:VWEHUCRS submitted 2026-07-22 math.DS

Chunlin Liu This is my paper · ORCID

classification math.DS MSC 37B0537B4037A3537A25

keywords Li–YorkechaosrelativemixingsoficentropyRokhlinPinskerfactorcountablegroupactionsscrambledCantorsetsprescribedsequences

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 proves that for actions of sofic groups, positive topological entropy is strong enough to guarantee Li–Yorke chaos not just somewhere, but along every infinite list of group elements chosen in advance. Chaos here means a Cantor set of points whose orbit segments at those prescribed times repeatedly come arbitrarily close together while also repeatedly separating by a fixed positive distance. The proof shows the real mechanism is relative mixing: any nontrivial relatively mixing factor extension already forces such scrambled Cantor sets on many fibers. This answers an open question from the literature and extends the conclusion to actions with positive Rokhlin entropy.

What carries the argument

The central object is a relatively mixing factor extension: for every pair of bounded functions with zero conditional expectation over Y, the conditional covariance of f and g·h tends to zero as g leaves every finite set. The key lemma shows that along any injective sequence (s_i), the averages (1/N)∑ U_{s_i} f converge to zero in L^2 for every centered f, and the convergence is simultaneous over a countable family of such functions. Combined with the Dirac-or-nonatomic dichotomy of conditional measures and the fact that relative mixing passes to the relatively independent square, this produces infinitely many close visits and infinitely many separated visits on almost every nonatomic fiber;

What would settle it

Exhibit a sofic group action with positive sofic topological entropy and one injective sequence (s_i) for which no pair of distinct points has liminf distance 0 and limsup distance larger than some fixed δ—equivalently, along s_i all orbit pairs either stay apart after some time or never separate. The theorem asserts this cannot happen; finding such an action, or showing the cited relative-mixing theorem fails for the relevant Pinsker factor, would refute the central claim.

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Extended reading notes

Core claim

The central discovery is a measure-theoretic theorem independent of entropy: if π:(X,μ,G)→(Y,ν,G) is a nontrivial relatively mixing extension of a compact metrizable G-space, then for every injective sequence (s_i) in G there is a Cantor set in X whose distinct points have liminf ρ(s_i x,s_i x')=0 and limsup ρ(s_i x,s_i x')>δ, with δ independent of the sequence (Theorem 1.3). The proof uses the relatively independent square over Y, a Hilbert-space averaging lemma showing that centered functions average to zero along any injective sequence, and a dichotomy that each conditional measure of a relatively mixing extension is either Dirac or nonatomic. On nonatomic fibers, a classical Cantor-set c

Load-bearing premise

The conclusive step for the sofic application is the cited theorem that every sofic measure-preserving action is relatively mixing over its sofic Pinsker factor; if that theorem does not hold under exactly the hypotheses used here, or if the non-isomorphic Pinsker factor it produces violates the freeness or ergodicity conditions the cited result requires, the proof of Theorem 1.1 collapses.

Editorial extensions

If this is right

Positive sofic topological entropy (with respect to any fixed sofic approximation) implies that for every injective sequence of group elements there is a Cantor scrambled set, with a fixed separation constant δ that does not depend on the chosen sequence.
The same conclusion holds for any countably infinite group for which the action has an essentially free invariant measure of positive Rokhlin entropy.
The method produces higher-order scrambled sets: for each r≥2, a Cantor set in which every r-tuple of distinct points has liminf max pairwise distance 0 and limsup min pairwise distance > δ_r.
It partially resolves the broader question about naive entropy: by a comparison theorem, positive sofic entropy implies positive naive entropy, so the chaotic conclusion holds under positive naive entropy in the sofic case; whether positive naive entropy alone suffices remains open.
The abstract criterion works for any nontrivial relatively mixing extension, not only entropy factors, so other sources of relative mixing yield the same prescribed-sequence chaos.

Reading between the lines

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

A natural inference: because Theorem 1.3 is independent of entropy, any pair of systems connected by a nontrivial relatively mixing factor—for instance a relatively mixing joining or an extension with relative spectral gap—should exhibit prescribed-sequence Li–Yorke chaos, so the entropy applications are only the first use. (editorial inference)
The uniform δ in Theorem 1.3(ii) suggests one can choose the chaotic Cantor sets from fibers in a fixed positive-measure family before the sequence is revealed; one might expect a full-measure-in-sequence version in which the same Cantor sets work for all sequences simultaneously, though the paper only asserts this fiber-family uniformity. (editorial inference)
The Dirac/non-atomic dichotomy implies that the nontriviality of the extension, not the size of the entropy, is what matters; actions with zero entropy but a nontrivial relatively mixing factor would still be chaotic along all sequences. This could be tested on classical zero-entropy relatively mixing extensions. (editorial inference)
The higher-order statement could be pushed to infinite unordered tuples if the finite-r constants stabilize or if a diagonal argument over r is possible; the paper does not carry this out. (editorial inference)

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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

Here's my take. The paper answers Huang–Li–Ye's question and does it with a genuinely new mechanism. Theorem 1.3 says any nontrivial relatively mixing extension gives Li–Yorke scrambled Cantor sets along every injective sequence, with a uniform separation constant. The proof is self-contained and I couldn't find a gap: relative mixing passes to the relative square, the Blum–Hanson averaging argument works for injective sequences, the Dirac/non-atomic dichotomy is clean, and the Mycielski step is standard. The sofic and Rokhlin applications are then straightforward, conditional on two cited results: Hayes's theorem that sofic p.m.p. actions are relatively mixing over their sofic Pinsker factors, and Seward's theorem on relative CPE+ for essentially free actions. I checked the statements in the paper and they match the cited theorems' formulations, but I did not verify Hayes's proof or whether his definition of Pinsker factor is exactly the maximal zero-entropy factor in the non-ergodic case. That's the soft spot: if Hayes's theorem has hidden hypotheses (ergodicity, freeness, a different Pinsker definition), Theorem 1.1 would need adjustment. The same goes for Seward in Theorem 1.4. This is not a flaw in the internal logic — the paper is honest about citing them — but a referee should confirm those theorem statements.

Minor: Proposition 4.1 (higher-order scrambled sets) is only sketched, but it's peripheral. The remark about naive entropy is speculative but flagged as such.

Overall: this is a serious paper, clearly written, answers an open question, and introduces a reusable tool. I'd send it to a good referee.

Referee Report

3 major / 4 minor

Summary. The paper proves a measure-theoretic criterion (Theorem 1.3): if a nontrivial extension of p.m.p. G-systems is relatively mixing, then for every injective sequence in G there exist fiberwise dense Mycielski sets along which distinct points have liminf of the observed metric equal to 0 and limsup bounded below by a positive constant; part (ii) makes the constant independent of the prescribed sequence. The proof combines relative mixing of the relatively independent square, a relative Blum–Hanson averaging lemma for injective sequences, a Dirac/non-atomic dichotomy for conditional measures, and a fiberwise Mycielski construction. The criterion is then applied in §5.1, using Hayes's theorem that the sofic Pinsker extension is relatively mixing, to deduce that positive topological sofic entropy implies Li–Yorke chaos along every injective sequence (Theorem 1.1), answering Huang–Li–Ye's question. In §5.2, using Seward's theorem on relative mixing of CPE+ extensions, the same conclusion is deduced for essentially free actions with positive Rokhlin entropy (Theorem 1.4). A higher-order scrambled version is sketched in §4.3.

Significance. If the external hypotheses are met, this is a substantial result: it answers an open question of Huang, Li, and Ye and extends the amenable-group theorem to the sofic setting through a mechanism that is conceptually independent of conditional-entropy additivity. The proof of Theorem 1.3 is self-contained, carefully executed, and has no free parameters: the averaging argument, the dichotomy, and the Mycielski construction are all explicit, and the separation constants are chosen before the sampling sequence, which is a strong feature. The main risk is not internal circularity but the precise applicability of the two quoted external theorems, which are the load-bearing links from the abstract criterion to the topological and Rokhlin-entropy conclusions.

major comments (3)

[§5.1, proof of Theorem 1.1] Theorem 1.1 depends entirely on Hayes's theorem [8, Theorem 3.4(i)] that the sofic Pinsker extension is relatively mixing. The measure μ obtained from the variational principle is an arbitrary invariant measure with h_{Σ,μ}>0; it is not shown to be ergodic, of finite sofic entropy, or to satisfy any other hypothesis that [8, Theorem 3.4(i)] may require. The manuscript only states 'By [8, Theorem 3.4(i)], the extension πΣ is relatively mixing' without stating the precise hypotheses of that theorem or verifying them for the non-ergodic, possible infinite-entropy measure appearing here. Since this is the key step that makes the main answer to Huang–Li–Ye hinge on an external result, the authors should either quote the theorem verbatim and prove its hypotheses hold, or supply a relative-mixing theorem for the exact class of measures produced by the variational principle.
[§5.2, proof of Theorem 1.4] Theorem 1.4 relies on Seward's [16, Corollary 5.2(1)], whose hypotheses are only partially stated. The proof asserts that the outer Rokhlin Pinsker factor gives a relative CPE+ extension and that essential freeness is enough, but it does not verify that Seward's set-theoretic relative mixing formulation is exactly equivalent to Definition 3.1 in this generality; Remark 3.2 asserts equivalence without proof. If Seward's theorem requires additional hypotheses (e.g., ergodicity, finite Rokhlin entropy, or a particular factorization of the Pinsker factor), Theorem 1.4 may not follow as stated. The authors should state Seward's theorem in full and give a direct verification, rather than relying only on a short assertion.
[§4.3, Proposition 4.1] Proposition 4.1 is the advertised higher-order extension, but its proof is only a sketch. The key step—applying Lemma 3.5 to the r-fold relatively independent joining and then repeating the Fatou/Diraction arguments to get simultaneous full-measure close and separated visits for r-tuples—is not written out. For a formal proposition, the reader needs to see the measurable-set bookkeeping (e.g., the analogues of D_n, V_n, and the positive-measure family from which n_r is chosen) and the higher-order Mycielski argument, especially since the abstract advertises higher-order scrambled Cantor sets. This is not load-bearing for Theorem 1.1, but it is part of the claimed contribution.

minor comments (4)

[Theorems 1.1 and 1.3] The theorem statements promise a Cantor set K_s, but the proof via Theorem 1.3 produces a Mycielski set M_{y,s}, i.e., a countable union of Cantor sets. Since any Cantor subset of M_{y,s} inherits the desired property, the extraction step should be stated explicitly (or the statements should say 'Cantor set' and add one sentence in §4.2).
[§4.3] In the sketch of Proposition 4.1, the assertion that μ_y^{⊗r}(B_{r,n}) ↗ 1 for non-atomic μ_y implicitly uses the fact that the set of tuples with two equal coordinates has zero μ_y^{⊗r}-measure; this should be mentioned. Also, 'Holder inequality' should be 'Hölder inequality'.
[Remark 3.2] The notation gB for the image of a Borel set under the homeomorphism g is fine, but the phrase 'with the convention gB={gx:x∈B}' should be stated before the displayed formula, because it is used in equation (2) and in the subsequent discussion.
[General] There are minor typographical spacing issues in the abstract and in the first lines of Section 1 (e.g., 'letπ:' should be 'let π:'). These do not affect content.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.3 is derived self-containedly from the definition of relative mixing, and the applications invoke independent external theorems rather than importing the conclusion.

full rationale

The paper's central abstract result, Theorem 1.3, is a self-contained derivation: starting from the definition of relative mixing (Definition 3.1), it proves relative mixing of the relatively independent square (Lemma 3.3), a relative Blum–Hanson averaging lemma (Lemma 3.4), a simultaneous-averaging lemma (Lemma 3.5), and a Dirac/non-atomic dichotomy for conditional measures (Lemma 3.6), then combines these with Mycielski's theorem. No fitted parameters are involved, and no step defines its conclusion into its hypothesis. The applications in Theorems 1.1 and 1.4 are conditional on external results by Hayes [8, Theorem 3.4(i)] and Seward [16, Corollary 5.2(1)] asserting relative mixing of the relevant Pinsker extensions; these are independent published theorems and do not assume the target Li–Yorke conclusion. The paper contains no self-citations of the author that bear on the derivation, and the cited uniqueness or Pinsker-factor facts are not invoked as replacements for proof in a circular way. The main risks are external: whether Hayes's and Seward's hypotheses, such as ergodicity or freeness, are satisfied by the measures and extensions used here. Those are correctness/verification concerns, not circularity. The proof sketch of Proposition 4.1 is admittedly brief but not circular, and no in-text limitation passage asserts a circular dependency.

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

The central proof of Theorem 1.3 is self-contained given standard measure theory and Mycielski's theorem; it introduces no fitted parameters and no invented entities. The applications rely on two deep external theorems (Hayes for sofic Pinsker factors, Seward for Rokhlin entropy) and the sofic variational principle, all cited but not reproved.

assumptions (6)

standard math Standard disintegration of μ over the factor Y and existence of conditional expectations.
Used throughout Sections 2.2 and 4 to define fiberwise measures p_m, q_n and conditional expectations; standard measure theory.
standard math Mycielski's theorem: a dense Gδ relation in a perfect compact metric space admits a dense Mycielski set whose off-diagonal pairs lie in the relation.
Invoked in Sections 4.2 and 4.3, cited as [15, Theorem 1].
standard math Characterization of relative weak mixing via relative ergodicity of the relatively independent self-joining.
Used in Lemma 3.6 to derive the Dirac/non-atomic dichotomy; cited to Glasner [7, Chapter 9, Section 5].
domain assumption Sofic variational principle: h^top_Σ(X,G)=sup_μ h_Σ,μ(X,G).
Used in the proof of Theorem 1.1 to obtain an invariant measure of positive sofic measure entropy; cited to Kerr–Li [12, Theorem 6.1].
domain assumption Hayes's theorem: every sofic p.m.p. action is relatively mixing over its sofic Pinsker factor.
Load-bearing for Theorem 1.1; invoked in Section 5.1 via [8, Theorem 3.4(i)].
domain assumption Seward's corollary: an essentially free relative-CPE+ extension over the outer Rokhlin Pinsker factor is relatively mixing.
Load-bearing for Theorem 1.4; invoked in Section 5.2 via [16, Corollary 5.2(1)].

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Cite this review

Pith. "Pith review of Li--Yorke Chaos Along Any Infinite Sequence: Relative Mixing, Sofic and Rokhlin Entropy." pith.science (2026). https://pith.science/paper/VWEHUCRS

@misc{pith2026260720735,
  author       = {Pith},
  title        = {Pith review of: Li--Yorke Chaos Along Any Infinite Sequence: Relative Mixing, Sofic and Rokhlin Entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWEHUCRS}},
  note         = {Machine review of arXiv:2607.20735}
}

abstract

Let $G$ be a countably infinite discrete group and let $\pi:(X,\mu,G)\to(Y,\nu,G)$ be a nontrivial relatively mixing extension, where $X$ is a compact metrizable $G$-space. We prove that there exists a constant $\delta>0$ such that, for every injective sequence $(s_i)_{i\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\subseteq X$ whose distinct points $x,x'$ satisfy \[ \liminf_{i\to\infty}\rho(s_i x,s_i x')=0, \qquad \limsup_{i\to\infty}\rho(s_i x,s_i x')>\delta. \] The method also yields higher-order scrambled Cantor sets. As a principal application, for a sofic group $G$, positive topological sofic entropy implies the preceding conclusion, answering a question of Huang, Li, and Ye. The same conclusion also holds for actions of arbitrary countably infinite discrete groups admitting an essentially free invariant measure of positive Rokhlin entropy.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

2, 239–246

Daniel Berend and Vitaly Bergelson,Mixing sequences in Hilbert spaces, Proceedings of the American Mathematical Society98(1986), no. 2, 239–246

1986

[2] [2]

Reine Angew

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2002

[3] [3]

Systems22(2002), no

François Blanchard, Bernard Host, and Sylvie Ruette,Asymptotic pairs in positive- entropy systems, Ergodic Theory Dynam. Systems22(2002), no. 3, 671–686

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Peter Burton,Naive entropy of dynamical systems, Israel J. Math.219(2017), no. 2, 637–659

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Ben Hayes,Mixing and spectral gap relative to Pinsker factors for sofic groups, Pro- ceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F. R. Jones’ 60th Birthday, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 46, Australian National University, Canberra, 2017, pp. 193–221

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