Rescaled topological entropy

C. Rojas; E. Rego; X. Wen

REVIEW 2 major objections 5 minor 19 references

Rescaled topological entropy

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rescaled topological entropy: a new invariant for vector fields with singularities.

desk verdict A new invariant for singular flows with a mostly sound proof package; the gaps are repairable, and the paper deserves refereeing. read the letter →

arxiv 2506.02383 v1 pith:KIDU6D3E submitted 2025-06-03 math.DS

E. Rego , C. Rojas , X. Wen This is my paper

classification math.DS MSC 37C1037B05

keywords rescaledtopologicalentropysingularvectorfieldsmetricperiodicorbitgrowthexpansiveflowssurfaceclosedmanifolds

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

On a closed manifold, a C1 vector field can have zero topological entropy and still display complicated behavior, particularly when singularities are present. The paper constructs a new nonnegative number, the rescaled topological entropy e*(X), designed to capture this hidden complexity. The number is finite for every C1 vector field, bounds the classical topological entropy and the rescaled metric entropy from above, agrees with topological entropy when the field has no singularities, and is positive on explicit surface fields whose topological entropy is zero. It is invariant under rescaled topological conjugacy and, for rescaling-expansive fields with dynamically isolated singularities, bounds the exponential growth rate of periodic orbits. A sympathetic reader should take the main contribution to be a single invariant that genuinely extends topological entropy to singular flows and answers a question about periodic-orbit growth for near-expansive flows.

What carries the argument

The working object is the rescaled dynamical ball B*(x,t,epsilon) = { y in M : sup_{0 <= s <= t} d(phi_s(x), phi_s(y)) / ||X(phi_s(x))|| < epsilon }, with the accompanying rescaled spanning and separating numbers. e*(X) is defined as lim_{epsilon->0} limsup_{t->infinity} (1/t) log R*(t,epsilon), where R*(t,epsilon) counts the minimum number of such balls needed to cover the set M_epsilon = { x : ||X(x)|| >= epsilon }; a compact-subset and separating-set version, e*(X) = sup_K lim_{epsilon->0} limsup_{t->infinity} (1/t) log S*(t,epsilon,K), makes the invariant computable. Two further mechanisms carry the proofs: a local norm-comparison lemma saying that points within a small rescaled distance have comparable speeds, which gives the rescaled balls a usable ball-doubling structure, and the finiteness estimate e*(X) <= 2dL for a Lipschitz constant L of X. The positivity criterion (Proposition 1) reduces e*(X) > 0 to the existence of a compact set away from the singularities containing exponentially many points that stay uniformly separated in the rescaled metric along long times, which is exactly what the torus and sphere examples verify.

What would settle it

Compute e*(X) for a fixed singular vector field using two bi-Lipschitz equivalent Riemannian metrics; if the two limits differ after letting epsilon go to zero, the assignment of e*(X) to the field is not well-defined.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the Theorem of Section 4: every C1 vector field X on a closed manifold carries a finite nonnegative number e*(X), called its rescaled topological entropy, defined through the growth of rescaled dynamical balls rather than ordinary dynamical balls. This number satisfies e*_mu(X) <= e*(X) for every Borel probability measure with mu(Sing(X)) = 0, satisfies e(X) <= e*(X) with equality when X is nonsingular, is positive for a constructed C-infinity vector field on the two-torus, is unchanged by rescaled topological conjugacy, and satisfies limsup_{t->infinity} (1/t) log v(t) <= e*(X) whenever X is rescaling expansive with dynamically isolated singular set. The torus example is the key evidence that the new invariant is a strict extension of topological entropy: the field has zero topological entropy, yet exponentially large separated sets with uniform positive rescaled separation force e*(X) > 0. The theorem also yields corollaries for k*-expansive fields and for generic C^r vector fields whose singularities are hyperbolic, all of which then satisfy the periodic-orbit growth bound.

Load-bearing premise

The load-bearing premise is that the number e*(X) is independent of the Riemannian metric chosen to define rescaled distances and speeds, since otherwise it would not be a number assigned to the vector field alone.

Editorial extensions

If this is right

For every C1 vector field, the finite number e*(X) is an upper bound for the classical topological entropy, so the new invariant never undercounts complexity.
For nonsingular fields e*(X) equals e(X); the invariant is therefore a genuine extension, and any difference e*(X) - e(X) must come from the singular set.
Surface flows, which all have topological entropy zero, can still have e*(X) > 0, so the invariant detects a form of complexity invisible to classical entropy.
For rescaling-expansive, k*-expansive, or generic C^r vector fields with dynamically isolated singular set, the exponential growth rate of periodic orbits is bounded above by e*(X), extending the classical expansive-flow inequality to singular flows.
Since e* is invariant under rescaled topological conjugacy, it can be used to distinguish singular flows that are not conjugate in that sense.

Reading between the lines

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

The paper proves that metric entropy never exceeds e*, but not the reverse direction; a natural next step would be to look for a measure that realizes e* as a rescaled metric entropy, turning the half-variational principle into a full one.
The examples place the extra complexity at the singular set; a testable extension would be to compute e* for flows with one hyperbolic singularity and ask whether e* = e holds there, which would clarify what singular structure is actually needed to make the invariant positive.
The periodic-orbit bound opens a numerical route: estimate e* by counting orbits with period at most T in singular flows and compare the slope to a direct spanning-set computation; disagreement would indicate that the bound is not sharp or that the hypotheses are not tight.

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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

Punchline: this paper introduces a usable invariant for singular flows and proves the kind of theorem package that makes you want the result to be true. It deserves refereeing, but it is not quite airtight as written: the definition's dependence on the Riemannian metric is never addressed, and the T^2 example contains a false norm identity. Both look repairable.

The genuinely new content is the periodic orbit upper bound for rescaling expansive flows with dynamically isolated singular set, and the explicit positive-entropy examples on T^2 and S^2. The definition of e*(X) itself was in Rego's thesis, which the authors acknowledge, but the surrounding structure—the half-variational inequality, equality for nonsingular fields, conjugacy invariance, and the finiteness bound—is put together cleanly. The spanning/separating machinery in Sections 2–3 is mostly standard and carefully done; Lemma 4's finiteness proof is solid, and the Bowen–Walters argument in Item (6) follows the original structure and appears correct.

The soft spots, in order of importance. First, metric independence: Definition 1 uses the Riemannian distance and norm, and the paper never proves e*(X) is invariant under changing the metric. This is a real gap, not just cosmetic, because the singular set makes the rescaling nontrivial. I think it is repairable: using Lemma 3, one reduces to compact K ⊂ M*, and on such K the rescaled balls under two metrics satisfy a uniform inclusion with constants that absorb into the ε limit. But the lemma needs to be written. Second, the T^2 example asserts ||X(φ_n(p))|| = e^{-n}; with the stated ρ, the flow from x = −2 spends its first unit of time in the region where ρ = 1, so the identity fails at n = 1 and is off by a constant for larger n. The separation argument can be fixed (bound the norm above by Ce^{−n}, handle small n using the s = 0 term), but as written it is an unsupported equality. Third, the citations in the Item (1) chain are swapped: the first equality uses Lemma 9, the inequality uses Lemma 8. Fourth, the rescaled metric entropy input comes from the unpublished preprint [16]; this is a legitimate input, but the paper should either state the needed properties or ensure the reader can access them.

None of this is load-bearing in the sense of circularity or fitted constants. The central claims are proved directly from definitions. The paper is for researchers in topological dynamics of flows with singularities; it will be useful once the gaps are closed. I would send it to a serious referee and expect revision, not rejection.

Referee Report

2 major / 5 minor

Summary. The paper introduces a numerical invariant e*(X) for C1 vector fields X on closed Riemannian manifolds, called rescaled topological entropy, defined as the epsilon-to-0 limit of exponential growth rates of minimal cardinalities of spanning sets for rescaled dynamical balls of radius epsilon times the norm of the vector field along the flow. The main theorem asserts six properties: upper bounds for rescaled metric entropy and ordinary topological entropy, equality with topological entropy for nonsingular fields, positivity for an explicit torus vector field, invariance under rescaled topological conjugacy, and an upper bound on the growth rate of periodic orbits for rescaling expansive flows with dynamically isolated singular set. The paper also proves finiteness, gives separating and localized equivalent definitions, and derives corollaries for k*-expansive and C^r-generic rescaling expansive flows.

Significance. If the theorem is correct, e*(X) is a natural extension of topological entropy that can detect complexity of surface flows, whose topological entropy vanishes by Young's theorem, and it provides a positive answer to Question 2 in a neighborhood of the expansive context. The paper's main structural contribution is the definition itself together with the equivalence results in Lemmas 3, 5, and 9 and the explicit finiteness bound e*(X)<=2dL. The proofs are mostly self-contained and do not rely on fitted parameters, and Item (6) yields a quantitative bound via separating sets. However, two load-bearing points need attention: the metric independence of the definition is not proved, and the torus example in Item (4) contains an incorrect computation.

major comments (2)

[Section 2.2, Definition 1 and Theorem] The definition of e*(X) uses the distance d and norm ||.|| of a chosen Riemannian metric, but the paper never proves that e*(X) is independent of that choice. This is load-bearing because the theorem claims a number assigned to any C1 vector field and Item (5) asserts invariance under rescaled topological conjugacy; if e* depends on the metric, both statements are not well-founded. A short lemma using bi-Lipschitz equivalence of Riemannian metrics (with a constant C>=1 such that C^{-1}d_g<=d_h<=C d_g and C^{-1}||v||_g<=||v||_h<=C||v||_g) would give B*_g(x,t,epsilon) subset B*_h(x,t,C^2 epsilon) and M_{epsilon,h} subset M_{epsilon/C,g}, hence R*_h(t,epsilon) <= R*_g(t,epsilon/C^2), and therefore e*_h(X)=e*_g(X). This argument should be added.
[Section 4, proof of Item (4)] The assertion that ||X(phi_n(p))||=e^{-n} for p in C={-2}x[0,4] is false with the stated rho. For p=(-2,y), phi_1(p)=(-1,y) and rho(-1)=1, so ||X(phi_1(p))||=1 and not e^{-1}; moreover phi_2(p)=(0,y) and rho(0)=0, so phi_2(p) is a singularity and the norm is 0. Consequently the displayed lower bound for d*_n(p,q) and the conclusion e*(X)>0 are not established. The example needs to be repaired, for instance by choosing a circle on which the speed decays exponentially without hitting Sing(X), and then verified.

minor comments (5)

[Section 4, Item (1) proof] The citations in the chain sup_mu e*_mu(X)=sup_mu sup_K e*_mu(X,K) and then <= sup_K e*(X,K) are interchanged: Lemma 9 gives the equality e*_mu(X)=sup_K e*_mu(X,K), and Lemma 8 gives the inequality sup_mu e*_mu(X,K) <= e*(X,K).
[Section 4, Item (3) proof] The proof writes R*(t,epsilon,delta) and 'rescaled (t,epsilon/m(X),delta)-spanning', but Definition 1 has no delta parameter; the intended statement is R*(t,epsilon) <= R(t,m(X)epsilon) after the change of variables.
[Section 1 and Theorem] The theorem states 'any C1 vector field' while Section 1 says 'Throughout, we assume X != 0'; the nonzero assumption should appear in the theorem statement or the identically zero case should be handled separately.
[Section 4, Item (6) proof] In the final inequalities there are typographical inconsistencies ('S*(s,alpha,M_alpha)' and 'S*(t,alpha,N_delta)'); these should read M_delta.
[Section 4, Item (4) example] The statement Sing(X)={0,2/3}x[0,4] assumes that the unspecified smooth interpolations of rho do not create additional zeros; the authors should either specify rho completely or note that the interpolation can be chosen to avoid extra zeros.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: e*(X) is proved from its definition and auxiliary lemmas; cited prior work supplies inputs, not the claimed conclusions.

full rationale

The central quantity e*(X) is introduced in Definition 1 as a Bowen-Dinaburg type limit of rescaled spanning cardinalities R*(t,epsilon). The Theorem's six properties are then derived within the paper rather than imported: Item (1) compares e*_mu(X) with e*(X) through compact-set localizations (Lemmas 3, 8, 9); Item (3) is a direct two-sided estimate against ordinary entropy using min ||X||; Item (4) is an explicit torus example whose positivity is verified via Proposition 1, itself proved from the separating-set characterization; and Items (5)-(6) are consequences of the same definitions. References to work by the same authors ([12], [13], [16], [17], [18]) are used as input theorems or precedents, not as placeholders for the theorem being proved. For instance, [16] defines the rescaled metric entropy that Item (1) bounds, [18] supplies Lemma 1 and a shadowing-type theorem used inside the proof of Item (6), and [13] is used only to derive Corollary 1 from the main theorem. None of these citations asserts the conclusion that e*(X) has the stated properties. The reader's concern that e*(X) may depend on the auxiliary Riemannian metric is a possible well-definedness gap, but it is not a circular reduction: nothing in the proof assumes the conclusion by defining e* in terms of itself or of the quantities it is claimed to bound. Therefore no specific circular step can be exhibited, steps is empty, and the circularity score is 0.

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

No empirical data or fitted constants are used. The auxiliary functions and constants in the explicit examples are existential choices for the constructions, not hidden degrees of freedom in the invariant. The axioms listed below are the unproved background results imported from the literature, several of which are authored or co-authored by the present authors.

assumptions (6)

standard math Bowen-Dinaburg formula for topological entropy of C1 flows (Equation (2.1)).
Imported from [5,7] and used to compare e(X) with rescaled spanning counts.
standard math Variational principle for topological entropy of flows (Theorem A in [14]).
Used in Item (2) to reduce e(X) to ergodic invariant measures with zero singularity mass.
domain assumption Rescaled norm comparability lemma (Lemma 1 in [18]) and distortion bound (Lemma 2.5 in [9]).
Imported from cited papers by the authors; controls ratios ||X(phi_s z)||/||X(z)|| and is used for spanning/separating equivalences and finiteness.
domain assumption Well-definedness and properties of rescaled metric entropy e*_mu from [16].
[16] is an unpublished preprint by co-author Wen; the definition of e*_mu and the property used in Item (1) are not reproduced here.
domain assumption Rescaling expansiveness theorem, specifically Theorem 1.1-(v) in [18].
Used in Item (6) to convert proximity along orbits into equality up to a bounded time shift.
standard math Kupka-Smale theorem: C^r generic vector fields have hyperbolic singularities.
Used to derive Corollary 2 that a generic singularity set is dynamically isolated.

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Pith. "Pith review of Rescaled topological entropy." pith.science (2026). https://pith.science/paper/KIDU6D3E

@misc{pith2026250602383,
  author       = {Pith},
  title        = {Pith review of: Rescaled topological entropy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KIDU6D3E}},
  note         = {Machine review of arXiv:2506.02383}
}

abstract

We prove that to any smooth vector field of a closed manifold it can be assigned a nonnegative number called {\em rescaled topological entropy} satisfying the following properties: it is an upper bound for both the topological entropy and the rescaled metric entropy \cite{ww}; coincides with the topological entropy for nonsingular vector fields; is positive for certain surface vector fields (in contrast to the topological entropy); is invariant under rescaled topological conjugacy; and serves as an upper bound for the growth rate of periodic orbits for rescaling expansive flows with dynamically isolated singular set. Therefore, the rescaled topological entropy bounds such growth rates for $C^r$-generic rescaling (or $k^*$) expansive vector fields on closed manifolds.

Figures

Figures reproduced from arXiv: 2506.02383 by the authors.

**Figure 2.** Graphs of ρ0 and ρ where γ = ρ ′ (a)(1 − a 2 ) > 0. Thus, (4.7) kX(ϕt(p))k = ρ(z(t))p 1 − z 2(t) ≤ κe−γt , ∀t ≥ 0, where κ = ρ(−δ) √ 1 − δ 2 > 0. Now take tn = γ −1n for n ∈ N. We have from (4.6) and (4.7) that d ∗ tn (p, q) > √ 2κ −1 (1 − a 2 )π, ∀n ∈ N, p 6= q ∈ En. Consequently, inf n∈N inf p6=p∈En d ∗ tn (p, q) > 0. On the other hand, card(En) = 2n so lim sup n→∞ 1 tn log card(En) = γ log 2 > 0. Since tn → ∞, we… view at source ↗

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Works this paper leans on

19 extracted references · 19 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [16]

W ang, Y., W en, X., A rescaled metric entropy of C1-vector ﬁelds with singularities, Preprint (2024)

work page 2024

[2] [1]

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

work page 1965

[3] [2]

Systems 39 (2019), no

Arbieto, A., Cordeiro, W., Paciﬁco, M.J., Continuum-wi se expansivity and entropy for ﬂows, Ergodic Theory Dynam. Systems 39 (2019), no. 5, 1190–1210

work page 2019

[4] [3]

Artigue, A., Rescaled expansivity and separating ﬂows, Disc. Cont. Dynam. Syst. 38 (2018), 4433–4447

work page 2018

[5] [4]

305 (2022), Paper No

Artigue, A., Cordeiro, W., Pac ´ ıﬁco, M.J., N-expansive ﬂows, Topology Appl. 305 (2022), Paper No. 107869, 15 pp

work page 2022

[6] [5]

Bowen, R., Entropy for group endomorphisms and homogene ous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. 18 E. REGO, C. ROJAS, AND X. WEN

work page 1971

[7] [6]

Bowen, R., W alters, P., Expansive one-parameter ﬂows. J. Diﬀerential Equations 12 (1972), 180–193

work page 1972

[8] [7]

Dinaburg, E.I., The relation between topological entro py and metric entropy, Dokl. Akad. Nauk SSSR 190, (1970) 19–22 (Soviet Math. Dokl. 11 (1969), 13–16)

work page 1970