Relative topological entropy and relative mean dimension of induced factors

Kairan Liu; Yixiao Qiao

REVIEW 2 major objections 4 minor 1 cited by

Relative topological entropy and relative mean dimension of induced factors

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read For amenable group actions, a factor map has zero relative entropy exactly when its induced factor map does, and positive relative entropy exactly when the induced factor has infinite relative mean dimension.

desk verdict The result is important and likely correct, but the written proof has two real gaps: a false empirical-representation lemma in Section 3 and an unproved relative Lindenstrauss-Weiss theorem used for the converses. read the letter →

arxiv 2511.18040 v2 pith:HCNQZXNL submitted 2025-11-22 math.DS

Kairan Liu , Yixiao Qiao This is my paper · ORCID

classification math.DS MSC 37B4037B99

keywords relativetopologicalentropymeandimensioninducedfactormapsamenablegroupactionscombinatorialindependenceIE-pairsprobabilitymeasurespacesFølnersequences

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 proves a dichotomy for factor maps between compact dynamical systems acted on by a countably infinite amenable group. If the original factor map has zero relative topological entropy—meaning its fibers are not exponentially complex—then the induced factor map on the space of Borel probability measures also has zero relative topological entropy. Conversely, if the original factor map has positive relative topological entropy, the induced factor map has infinite relative mean dimension, so its fibers contain arbitrarily high-dimensional structure. Together these statements say that the relative complexity of the original map is completely mirrored in the induced measure system: the zero-versus-positive entropy threshold becomes a zero-versus-infinite dimension threshold. This matters because it converts a counting question into a geometric one and gives a new way to detect positive relative entropy through infinite-dimensional fibers.

What carries the argument

Combinatorial independence sets for factor maps. For a pair of sets A_1,A_2, a subset I of the group is an independence set if every finite assignment of group elements to A_1 or A_2 can be realized by a single point in a common fiber of π. Positive independence density—having such sets of size at least c|H| inside every finite H—is equivalent to positive relative entropy by a known characterization. Section 3 converts independence for measure-open sets in M(X) into independence for point-open sets in X via an averaging lemma that builds, for each binary coloring of a large group subset, a common fiber point. Section 4 turns an independence set of positive density into a continuous affine in

What would settle it

Construct a factor map π between two G-systems with G a countably infinite amenable group such that h_top(π,G)=0 but mdim(π,G)>0. Such a map would violate the contrapositive of Theorem 2.4 and therefore the converse direction of Theorem 1.1(ii); the paper's one-way implication (positive entropy yields infinite relative mean dimension of the induced factor) would still hold. Conversely, any example with h_top(π,G)>0 and mdim(eπ,G)<∞ would directly refute Theorem 1.1(ii).

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for a countably infinite amenable group G and a factor map π:(X,G)→(Y,G), the induced factor map eπ:(M(X),G)→(M(Y),G) on Borel probability measures satisfies h_top(π,G)=0 if and only if h_top(eπ,G)=0, and h_top(π,G)>0 if and only if mdim(eπ,G)=+∞. Relative topological entropy counts, per group element, how many orbit segments are needed to separate points within a single fiber; relative mean dimension counts, per group element, how many real parameters are needed to describe a fiber. The paper proves the forward directions directly: an independence set witnessing positive relative entropy of eπ is converted through an averaging argument into an independence

Load-bearing premise

The load-bearing premise is the unproved Theorem 2.4—that any factor map with finite relative topological entropy has zero relative mean dimension; the paper says the proof is 'almost the same' as the absolute case but omits it, and the cited theorem it is modelled on covers only the absolute, non-relative setting. If that relative statement fails, the 'only if' halves of Theorem 1.1 collapse while the proved one-way implications survive.

Editorial extensions

If this is right

Zero relative topological entropy of a factor map between amenable group actions is completely determined by the induced factor map on probability measures: one is zero exactly when the other is.
Positive relative topological entropy of the original map manifests at the induced level as infinite relative mean dimension—a geometric form of complexity, not merely infinite entropy.
Combined with the bridge theorem that finite relative entropy forces zero relative mean dimension, the result implies that in this setting a factor map cannot have both finite positive relative entropy and positive relative mean dimension.
In the absolute case where the factor is a single point, the theorem recovers and unifies known results: a system has zero topological entropy iff its induced system does, and positive entropy iff its induced system has infinite mean dimension.

Reading between the lines

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

The omitted proof of the bridge theorem is the delicate point. If a relative version of 'finite entropy implies zero mean dimension' fails for amenable group actions, the iff statements in Theorem 1.1 would reduce to one-way implications; a concrete falsifier would be a factor map with zero relative entropy and positive relative mean dimension.
The proof uses only independence density and a Lebesgue lemma, so the same scheme could likely be adapted to other induced spaces—such as the space of closed subsets or the measure center of a system—to obtain analogous entropy-versus-dimension dichotomies.
The construction in Section 4 is quantitative: the order of refining covers grows like 2^H while the group size grows polynomially in H, suggesting that when positive relative entropy is present, the relative mean dimension of the induced factor is not merely infinite but diverges at a controlled rate tied to the independence density.

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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

Main thing to know: this paper proves a relative version of two known induced-system phenomena, but the proof as written is not complete. The zero-entropy preservation step rests on a lemma (3.1) that is false as stated, and the 'if' directions of the main theorem rely on Theorem 2.4, which is stated without proof. Both look repairable, and the main ideas are likely sound, but a referee will need to force the repairs.

What's new: the result itself, for amenable group actions, is a meaningful extension of Glasner-Weiss (zero entropy of induced systems) and Burguet-Shi / Shi-Zhang (positive entropy implies infinite mean dimension of induced systems) from absolute systems to factor maps. The proof machinery—using IE-pairs and a Sauer-Shelah-type lemma to transfer independence from the induced system back to the original—is a real step forward, and Section 4's simplex embedding argument is an elegant way to get the huge fiber complexity.

The main proof has two soft spots, in proportion to how soft they are. First, the stress-test is right about Lemma 3.1. In this paper, M_n(X) is the set of uniform empirical measures of length n, so R_{eπ} is not the union of the R_{eπ_n}; the union is only the empirical pairs. The proof needs a simultaneous approximation argument (choose L large, approximate all λ_σ and ν by common empirical averages with small error), and the strict inequalities would then be preserved by choosing the slack carefully. As written, it's a gap, though not one that obviously sinks the result. Second, Theorem 2.4 (finite relative entropy implies zero relative mean dimension) is simply asserted, and it's load-bearing for the 'if' directions. If the proof is really 'almost the same' as Lindenstrauss-Weiss, the authors should write it out, because the relative amenable-group setting is not identical.

The other direction—positive entropy of π forces infinite relative mean dimension of eπ—looks solid modulo the usual checks. Claim 3's use of the Burguet-Shi Lebesgue lemma is the right tool.

My overall take: this is a serious paper with a likelier-than-not correct theorem, but it is not ready in its current form. The gaps are fixable and the combinatorial core is plausible. I'd send it to a competent referee—not desk reject it. The referee should ask for a corrected treatment of the empirical approximation step and a full proof of Theorem 2.4.

Referee Report

2 major / 4 minor

Summary. The paper studies the relation between relative topological entropy and relative mean dimension for factor maps of actions of countably infinite amenable groups, and the corresponding induced factor maps on spaces of Borel probability measures. Theorem 1.1 claims: (i) a factor map π has zero relative topological entropy if and only if the induced factor map eπ does; (ii) π has positive relative topological entropy if and only if eπ has infinite relative mean dimension. The proof strategy is to prove the two hard one-way directions — htop(π)=0 ⇒ htop(eπ)=0 in Section 3 and htop(π)>0 ⇒ mdim(eπ)=∞ in Section 4 — and then to use a relative version of the Lindenstrauss–Weiss theorem (Theorem 2.4) to obtain the converses. Section 3 uses relative independence sets and a Sauer–Shelah-type combinatorial lemma; Section 4 builds large simplexes inside fibers of eπ and applies a tailored Lebesgue lemma. The conclusions are strong and the overall architecture is coherent, but two load-bearing ingredients are problematic as written.

Significance. If the gaps are repaired, this is a significant contribution. The paper establishes a complete quantitative dictionary between fiber complexity of a factor map and that of its induced measure factor: zero relative entropy is preserved exactly, and positive relative entropy forces infinite relative mean dimension in the induced system. This extends the Glasner–Weiss and Burguet–Shi results to relative entropy/mean dimension for amenable group actions, and the independence-based method in Section 3 is a novel route. The construction in Section 4, embedding high-dimensional simplexes in fibers of the induced map, is elegant. The paper also makes good use of machine-checkable-style combinatorial lemmas and gives a self-contained proof of the main combinatorial ingredient (Lemma 3.2). However, the current manuscript contains an unproved theorem and a false lemma that are essential to the main theorem, so the central claims are not yet established as written.

major comments (2)

[Section 3, Lemma 3.1 and Eq. (7)] Lemma 3.1 as stated is false. If X=Y=[0,1] and π=id, then R_{eπ} is the diagonal of M(X)×M(X); taking ν to be Lebesgue measure gives (ν,ν)∈R_{eπ}, but ν is not in any M_n(X), since M_n(X) consists of uniform empirical measures on n points. Thus R_{eπ} is not the union of the R_{eπ_n}. At most one has a density statement, R_{eπ} = closure(⋃_n R_{eπ_n}). This matters because Proposition 3.4 uses Lemma 3.1 to pass from (6) to the exact representation (7), with a common L and exact equalities ν=(1/L)Σδ_{y_i} and λ_σ=(1/L)Σδ_{x_i^σ}. Without a simultaneous approximation argument, the strict inequalities in (6) and the subsequent counting argument leading to the independence set for (A_1,A_2) are not justified. The forward direction htop(π)=0 ⇒ htop(eπ)=0 is therefore incomplete as written.
[Section 2, Theorem 2.4] Theorem 2.4 states that finite relative topological entropy implies zero relative mean dimension for a factor map of amenable group actions. The proof is omitted with the comment that it is 'almost the same' as [LW00, Theorem 4.2]. However, [LW00, Theorem 4.2] is the absolute case, and the relative version is load-bearing: after the one-way implications in Sections 3 and 4, Theorem 2.4 is exactly what converts them into the if-and-only-if statements of Theorem 1.1. The authors should either give a full proof or cite a reference that contains the relative amenable-group version. As it stands, the converses of both (i) and (ii) rest on an unproved assertion.

minor comments (4)

[References] Several entries in the bibliography appear not to be cited in the text: [Dow11], [GTW00], [Lia22]. These should be removed or cited.
[Page 11, Claim 1 proof] Typo: 'assuem' should be 'assume'.
[Section 4, definition of Ψ] The injectivity of Ψ is used implicitly when identifying L_n with Δ_{[2]^H}^{m_n}. It is true because the sets defining x_E are pairwise disjoint for distinct E, but it should be stated explicitly.
[Section 4, construction of h_{i,j}] The sentence 'Take h_{i,j} ∈ {h_ℓ : ℓ∈[M]} such that h_{i,j} ≠ h_{i,j'} for any i∈[m_n] and distinct j,j′∈[H]' could be misread as requiring distinctness across different i. It is clearer to say 'for each fixed i, the elements h_{i,1},...,h_{i,H} are pairwise distinct'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hard one-way directions are genuine transfer/construction arguments, and the main equivalences do not reduce to their inputs by definition; the noted omitted proof and doubtful cited lemma are correctness risks, not circularity.

full rationale

The paper's central claim is not assumed. Section 3 proves htop(π)=0 ⇒ htop(eπ)=0 by contraposition: it assumes htop(eπ)>0, obtains a nontrivial IE-pair (μ1,μ2) via Theorem 2.1 (cited from [LW24]), separates them by disjoint open sets A1,A2, and uses an independence set for (U1,U2) with respect to eπ plus Lemma 3.1 to transfer independence to (A1,A2) with respect to π. This is a substantive transfer argument: the conclusion is not one of the hypotheses, and the auxiliary constants a1,a2,b,d,τ are chosen to satisfy inequalities, not fitted to the target. Section 4 proves htop(π)>0 ⇒ mdim(eπ)=∞ by constructing affine images of high-dimensional simplices inside fibers of eπ over δ_y, using positive independence density and the Burguet–Shi Lebesgue lemma; the large simplex dimension is derived from the independence density parameter r rather than imported from mdim. The reverse directions are explicitly reduced to Theorem 2.4: 'Note that by Theorem 2.4, the statements htop(π,G)=0⇔htop(eπ,G)=0 and htop(π,G)>0⇔mdim(eπ,G)=+∞ of Theorem 1.1 reduce to the derivations...'. Theorem 2.4 is stated with 'the proof is omitted as is (almost) the same' and is a relative generalization of [LW00, Theorem 4.2]; this is an omitted-proof / correctness risk, not circularity, because the theorem is not the target result and is not derived from it. Likewise, the cited Lemma 3.1 ([LW24, Lemma 4.1]) is used in Section 3 and appears false as stated for non-empirical measures; that is an invalidity concern in the proof, not a self-definitional reduction. Self-citations such as [LW24] and [LL25] are present, but they supply independent tools and the central claim does not reduce to them by construction. Therefore the derivation chain is not circular.

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

No empirical free parameters; proof constants are auxiliary. The paper relies on external deep results, two from the first author's prior work, but not on assumptions that include the target theorem.

assumptions (6)

domain assumption Theorem 2.1: htop(π,G)>0 iff IE(π,G)\Δ2(X)≠∅ iff disjoint closed sets with positive independence density exist.
Borrowed from [LW24]; used to detect positive relative entropy in both directions. It is a deep local-entropy result not proved here.
domain assumption Theorem 2.4: finite relative topological entropy implies zero relative mean dimension.
Stated in Section 2 with proof omitted; used to reduce Theorem 1.1 to the two one-way implications. If false, the iff characterizations fail.
domain assumption Lemma 3.1: R_{eπ}=closure of ∪_n R_{eπ,n}.
Borrowed from [LW24, Lemma 4.1]; used to approximate fiber measures by empirical measures in Proposition 3.4.
domain assumption Lemma 2.5: Burguet-Shi tailored Lebesgue lemma.
Borrowed from [BS25, Lemma 9]; provides the key lower bound for order of covers in Section 4.
standard math Karpovsky-Mailman / Sauer-Perles-Shelah lemma.
Cited [KL16, Lemma 12.14]; used in Lemma 3.2 to obtain a shattered subset I from a large family of colorings.
standard math Independence of relative entropy and relative mean dimension from Følner sequences and compatible metrics.
Invoked in the definitions; standard facts from [KL16, Theorem 4.38] and prior literature.

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Pith. "Pith review of Relative topological entropy and relative mean dimension of induced factors." pith.science (2026). https://pith.science/paper/HCNQZXNL

@misc{pith2026251118040,
  author       = {Pith},
  title        = {Pith review of: Relative topological entropy and relative mean dimension of induced factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCNQZXNL}},
  note         = {Machine review of arXiv:2511.18040}
}

read the original abstract

We study the relation of relative topological entropy and relative mean dimension between a factor map and its induced factor map for amenable group actions. On the one hand, we prove that a factor map has zero relative topological entropy if and only if so does the induced factor map. On the other hand, we prove that a factor map has positive relative topological entropy if and only if the induced factor map has infinite relative mean dimension.

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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

Entropies of compact subsets and supported measures
math.DS 2026-08 conditional novelty 7.0 of 10

For a compact subset K of a topological dynamical system, upper capacity entropy of K is zero exactly when the induced measure system on M(K) has zero entropy, otherwise the latter is infinite; packing entropies vanis...

Reference graph

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

R. L. Adler, A. G. Konheim, M. H. McAndrew. Topological entropy. Trans. Amer. Math. Soc. 114(1965) 309--319

1965

[2] [2]

Bauer, K

W. Bauer, K. Sigmund. Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79(1975) 81--92

1975

[3] [3]

R. Bowen. Erratum to ``Entropy for group endomorphisms and homogeneous space''. Trans. Amer. Math. Soc. 181(1973) 509--510

1973

[4] [4]

Burguet, R

D. Burguet, R. Shi. Topological mean dimension of induced systems. Trans. Amer. Math. Soc. 378(2025) 3085--3103

2025

[5] [5]

D. Dou, W. Huang, K.K. Park. Entropy dimension of topological dynamical systems. Trans. Amer. Math. Soc. 363(2011) 659--680

2011

[6] [6]

D. Dou, W. Huang, K.K. Park. Entropy dimension of measure preserving systems. Trans. Amer. Math. Soc. 371(2019) 7029--7065

2019

[7] [7]

Downarowicz

T. Downarowicz. Entropy in dynamical systems. Cambridge University Press (2011)

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[8] [8]

Glasner, J.P

E. Glasner, J.P. Thouvenot, B. Weiss. Entropy theory without a past. Ergodic Theory Dynam. Systems 20(2000) 1355--1370

2000