Polytopes and $C^0$-Riemannian metrics with positive $h_{\rm top}$

Marcelo R. R. Alves; Matthias Meiwes

arxiv: 2602.08521 · v2 · submitted 2026-02-09 · 🧮 math.DS · math.SG

Polytopes and C⁰-Riemannian metrics with positive h_(rm top)

Marcelo R. R. Alves , Matthias Meiwes This is my paper

Pith reviewed 2026-05-16 05:56 UTC · model grok-4.3

classification 🧮 math.DS math.SG

keywords Reeb flowstopological entropystarshaped polytopesC0-stabilitygeodesic flowsRiemannian metricscontact dynamicssmoothings

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The pith

Starshaped polytopes exist whose every smoothing yields Reeb flows with positive topological entropy

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that certain starshaped polytopes in four-dimensional space have the property that every starshaped smoothing of their boundary produces a Reeb flow with positive topological entropy. This settles an open question of Ostrover and Ginzburg by exhibiting examples where the entropy is robust under smoothing. The argument applies a C0-stability theorem for positive entropy of three-dimensional Reeb flows to these polytopal boundaries. A parallel construction yields continuous non-differentiable Riemannian metrics on any closed surface whose geodesic flows retain arbitrarily high topological entropy under every smoothing. These results show how non-smooth geometric objects can enforce persistent dynamical complexity.

Core claim

There exist starshaped polytopes P such that for any starshaped smoothing of ∂P the associated Reeb flows have positive topological entropy. This answers a question of Ostrover and Ginzburg. Similarly, given a closed surface M and C>0, there exist continuous and non-differentiable Riemannian metrics g on M with h_top>C in the sense that for any smoothing of g the associated geodesic flows have h_top>C.

What carries the argument

The C^0-stability of positive topological entropy for Reeb flows in dimension three, applied to starshaped smoothings of polytopes.

If this is right

Such polytopes provide examples where positive entropy persists under all starshaped smoothings of the boundary.
The construction affirmatively resolves the question of Ostrover and Ginzburg.
Parallel arguments produce C^0 Riemannian metrics on closed surfaces with arbitrarily large entropy bounds that survive all smoothings.
Topological entropy remains positive for the associated geodesic flows of these metrics under smoothing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar polytopal constructions could extend to higher-dimensional contact manifolds if the stability result generalizes.
Positive entropy may turn out to be C^0-open within suitable classes of Reeb flows on contact three-manifolds.
Concrete polytopes such as smoothed simplices could be checked numerically to confirm entropy values in specific cases.
These robust examples may inform minimal-entropy questions in Reeb dynamics and contact geometry.

Load-bearing premise

The C^0-stability of positive topological entropy for Reeb flows in dimension three holds and applies directly to the smoothings of the polytopes considered here.

What would settle it

An explicit construction of even one starshaped smoothing of a candidate polytope whose Reeb flow has zero topological entropy would disprove the claim.

read the original abstract

We study Reeb dynamics on starshaped hypersurfaces in $\mathbb{R}^4$ arising as smoothings of starshaped polytopes. Using the $C^0$--stability of positive topological entropy for Reeb flows in dimension three from our joint work with Dahinden and Pirnapasov, we show that there exist starshaped polytopes $P$ such that for any starshaped smoothing of $\partial P$ the associated Reeb flows have positive topological entropy. This answers a question of Ostrover and Ginzburg. Similarly, we show that given a closed surface $M$ and a number $C>0$, there exist continuous and non-differentiable Riemannian metrics $g$ on $S$ with $h_{\rm top}>C$ in the sense that for any smoothing of $g$ the associated geodesic flows have $h_{\rm top}>C$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper settles an open question on positive entropy for Reeb flows on smoothed starshaped polytopes by applying a prior C0-stability result, with a parallel statement for high-entropy Riemannian metrics.

read the letter

The key point is that the authors find starshaped polytopes in R^4 such that every starshaped smoothing of the boundary gives a Reeb flow with positive topological entropy. This answers the Ostrover-Ginzburg question directly. They also produce continuous Riemannian metrics on surfaces where any smoothing keeps topological entropy above a fixed positive number. Both results come from feeding suitable examples into their earlier joint stability theorem with Dahinden and Pirnapasov for C0-close Reeb flows in dimension three. The polytopal construction is the concrete new piece: it shows the stability property can be made uniform over an entire family of smoothings rather than just one flow. The metric side is a similar robustness statement in a different geometric setting. Both are legitimate applications that were not already in the literature. The argument is short and focused once the prior theorem is granted. The main limitation is that everything rests on that earlier stability result holding verbatim for the smoothings considered here. The paper needs to confirm that arbitrary starshaped smoothings stay inside the precise class where the theorem applies and that the entropy lower bound is forced rather than merely preserved. Without explicit polytopes or a short verification step in the text, a reader has to trust that the construction works for every possible rounding of the corners. That check is doable but should be spelled out clearly. This is aimed at specialists in Reeb dynamics and contact geometry who already know the stability theorem. It is not a broad survey or a new method, but it gives a clean existence result for a question that was open. The work is coherent on its own terms and deserves a serious referee who can check the application details against the prior paper.

Referee Report

2 major / 2 minor

Summary. The paper constructs specific starshaped polytopes P in R^4 such that every starshaped smoothing of ∂P yields a Reeb flow with positive topological entropy, by direct application of the C^0-stability theorem for positive h_top from the authors' prior joint work with Dahinden and Pirnapasov. It likewise produces continuous non-differentiable Riemannian metrics g on any closed surface M with h_top(g) > C for arbitrary C > 0, such that every smoothing of g has geodesic flow entropy exceeding C. The claims answer a question of Ostrover and Ginzburg.

Significance. If the application of the prior stability result is justified for the constructed polytopes and metrics, the work supplies an affirmative resolution to the Ostrover-Ginzburg question by exhibiting explicit geometric objects whose entropy positivity is robust under all starshaped/C^0 smoothings. This demonstrates that positive topological entropy can be made stable in the C^0 topology for Reeb and geodesic flows in low dimensions, providing a concrete class of examples where entropy persists without requiring differentiability.

major comments (2)

[construction of polytopes and application of stability result] The central existence claim rests on applying the C^0-stability theorem from the authors' prior work directly to arbitrary starshaped smoothings of the constructed polytopes P. The manuscript does not contain an explicit verification (e.g., in the construction section or the proof of the main theorem) that every such smoothing remains C^0-close to a reference flow with positive entropy while preserving the contact-type and starshaped hypotheses required by that theorem. This check is load-bearing for the statement that h_top > 0 holds for ANY smoothing.
[Riemannian metrics construction] The analogous claim for C^0-Riemannian metrics on surfaces (second part of the main theorem) similarly invokes stability of geodesic-flow entropy under smoothing but provides no explicit argument that the chosen metrics ensure the smoothed flows stay inside the class covered by the prior result while achieving the lower bound C. Without this, the quantitative statement h_top > C for every smoothing is not fully supported.

minor comments (2)

[abstract] The abstract refers to 'our joint work with Dahinden and Pirnapasov' without a full bibliographic entry; the reference list should include the precise citation for reader convenience.
[introduction] Notation for the topological entropy h_top is used without an initial definition or reminder of its normalization (e.g., with respect to the contact form or Riemannian metric); a brief clarification in the introduction would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment of its significance in resolving the Ostrover-Ginzburg question. We address the major comments below and will revise the manuscript to incorporate explicit verifications as requested.

read point-by-point responses

Referee: The central existence claim rests on applying the C^0-stability theorem directly to arbitrary starshaped smoothings of the constructed polytopes P. The manuscript does not contain an explicit verification that every such smoothing remains C^0-close to a reference flow with positive entropy while preserving the contact-type and starshaped hypotheses.

Authors: We acknowledge that an explicit verification of the stability theorem's hypotheses is not detailed in the current version. The polytopes are constructed so that their boundaries are C^0-close to a reference hypersurface with positive entropy Reeb flow, and starshaped smoothings preserve the contact-type condition by definition. In the revised manuscript we will add a dedicated paragraph in the construction section (likely Section 3) that explicitly checks C^0-closeness in the C^0 topology on the space of hypersurfaces and confirms that the starshaped property is inherited, thereby justifying direct application of the theorem from our prior work with Dahinden and Pirnapasov. This will make the argument fully transparent. revision: yes
Referee: The analogous claim for C^0-Riemannian metrics on surfaces invokes stability of geodesic-flow entropy under smoothing but provides no explicit argument that the smoothed flows stay inside the class covered by the prior result while achieving the lower bound C.

Authors: We agree that the current text lacks an explicit verification for the metric case. The continuous metrics are built by C^0-approximating a smooth metric whose geodesic flow has entropy exceeding C, ensuring that any smoothing remains sufficiently close in the C^0 topology on metrics. In the revision we will insert a short subsection detailing the smoothing procedure, verifying that the geodesic flows of the smoothed metrics remain in the class to which the stability theorem applies, and confirming that the entropy lower bound C is preserved uniformly. This will support the quantitative statement rigorously. revision: yes

Circularity Check

1 steps flagged

Central claim depends on self-cited C^0-stability theorem applying to all starshaped smoothings of constructed polytopes

specific steps

self citation load bearing [Abstract]
"Using the $C^0$--stability of positive topological entropy for Reeb flows in dimension three from our joint work with Dahinden and Pirnapasov, we show that there exist starshaped polytopes $P$ such that for any starshaped smoothing of ∂P the associated Reeb flows have positive topological entropy."

The headline existence statement is justified solely by citing the authors' overlapping prior work and asserting that the new polytopes and their smoothings fall under that theorem. The manuscript provides no independent verification or explicit verification that every starshaped smoothing satisfies the C^0-closeness and Reeb-flow hypotheses of the cited stability result, so the positive-entropy conclusion is carried by the self-citation.

full rationale

The paper's main existence result for polytopes with positive topological entropy under any starshaped smoothing is obtained by direct application of a C^0-stability theorem from the authors' prior joint work with Dahinden and Pirnapasov. The construction of the polytopes provides independent geometric content, but the entropy conclusion reduces to invoking that prior result without an explicit check in the manuscript that arbitrary smoothings preserve the exact hypotheses (C^0-closeness and contact-type conditions). This qualifies as self-citation load-bearing per the enumerated patterns but does not reduce the derivation to a tautology or fitted input by construction; the prior theorem is treated as an external input. No other circular steps (self-definitional, ansatz smuggling, or renaming) are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the C^0-stability theorem from the authors' prior work; no free parameters or new invented entities are introduced in the abstract. The polytopes themselves are not constructed explicitly here.

axioms (1)

domain assumption C^0-stability of positive topological entropy for Reeb flows on 3-dimensional contact manifolds
Invoked directly from the authors' joint work with Dahinden and Pirnapasov to transfer positive entropy from a reference flow to all nearby C^0 perturbations arising from smoothings.

pith-pipeline@v0.9.0 · 5460 in / 1445 out tokens · 33572 ms · 2026-05-16T05:56:11.334103+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using the C^0–stability of positive topological entropy for Reeb flows in dimension three from our joint work with Dahinden and Pirnapasov, we show that there exist starshaped polytopes P such that for any starshaped smoothing of ∂P the associated Reeb flows have positive topological entropy.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.1. Let (M_j) be a starshaped smoothing of ∂P. Define h_top((M_j)) := lim inf h_top(M_j)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Alves, L

M.R.R. Alves, L. Dahinden, M. Meiwes, and L. Merlin.C 0-robustness of topological entropy for geodesic flows.J. Fixed Point Theory Appl., 24(2):42, 2022

work page 2022
[2]

Alves, L

M.R.R. Alves, L. Dahinden, M. Meiwes, and A. Pirnapasov.C 0-stability of topolog- ical entropy for Reeb flows in dimension3. Published inJournal of the European Mathematical Society: online first

work page
[3]

Alves and M

M.R.R. Alves and M. Meiwes. Braid stability and the Hofer metric.Annales Henri Lebesgue, 7:521–581, 2024

work page 2024
[4]

Buhovsky, V

L. Buhovsky, V . Humili`ere, and S. Seyfaddini. The action spectrum andC 0 symplec- tic topology.Math. Ann., 380(1-2):293–316, 2021

work page 2021
[5]

Chaidez and M

J. Chaidez and M. Hutchings. Computing Reeb dynamics on four-dimensional con- vex polytopes.J. Comput. Dyn., 8(4):403–445, 2021

work page 2021
[6]

C ¸ ineli, V

E. C ¸ ineli, V . L. Ginzburg, B. Z. G¨urel, and M. Mazzucchelli. On the barcode entropy of Reeb flows.Selecta Math. (New Ser.), 2025. To appear

work page 2025
[7]

Colin, P

V . Colin, P. Dehornoy, U. Hryniewicz, and A. Rechtman. Generic properties of 3-dimensional Reeb flows: Birkhoff sections and entropy.Comment. Math. Helv., 99(3):557–611, 2024

work page 2024
[8]

Geiges.An introduction to contact topology, volume 109 ofCambridge Studies in Advanced Mathematics

H. Geiges.An introduction to contact topology, volume 109 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008

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Ginzburg, B.Z

V .L. Ginzburg, B.Z. Gurel, and M. Mazzucchelli. Barcode entropy of geodesic flows. Published in Journal of the European Mathematical Society: online first., 2024

work page 2024
[10]

Haim-Kislev

P. Haim-Kislev. On the symplectic size of convex polytopes.Geom. Funct. Anal., 29(2):440–463, 2019

work page 2019
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Haim-Kislev, R

P. Haim-Kislev, R. Hind, and Y . Ostrover. On the existence of symplectic barriers. Selecta Math. (N.S.), 30(4):65, 2024

work page 2024
[12]

Haim-Kislev and Y

P. Haim-Kislev and Y . Ostrover. A counterexample to Viterbo’s conjecture.Ann. of Math. (2). To appear

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Humili`ere, R

V . Humili`ere, R. Leclercq, and S. Seyfaddini. Coisotropic rigidity andc 0-symplectic geometry.Duke Math. J., 164(4):767–799, 2015

work page 2015
[14]

A. Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes ´Etudes Sci. Publ. Math., (51):137–173, 1980. 10 MARCELO R.R. ALVES AND MATTHIAS MEIWES

work page 1980
[15]

Katok and B

A. Katok and B. Hasselblatt.Introduction to the modern theory of dynamical systems, volume 54 ofEncyclopedia of Mathematics and its Applications. Cambridge Univer- sity Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza

work page 1995
[16]

M. Meiwes. Topological entropy and orbit growth in link complements.Math. Ann., 393(1):617–665, 2025

work page 2025
[17]

Z. Nitecki. On semi-stability for diffeomorphisms.Invent. Math., 14:83–122, 1971

work page 1971
[18]

Opshtein.C 0-rigidity of characteristics in symplectic geometry.Ann

E. Opshtein.C 0-rigidity of characteristics in symplectic geometry.Ann. Sci. ´Ec. Norm. Sup´er. (4), 42(5):857–864, 2009

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Paternain.Geodesic flows, volume 180 ofProgress in Mathematics

G.P. Paternain.Geodesic flows, volume 180 ofProgress in Mathematics. Birkh ¨auser Boston, Inc., Boston, MA, 1999

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R. T. Rockafellar. Convex analysis:(pms-28). 2015

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Tsodikovich

D. Tsodikovich. Hamiltonian dynamics on simplexes. Master’s thesis, Tel Aviv Uni- versity, 2018. MARCELOR.R. ALVES, INSTITUTE OFMATHEMATICS,, UNIVERSITY OFAUGSBURG, CHAIRANALYSIS ANDGEOMETRY, UNIVERSIT ¨ATSSTRASSE14, DE-86159 AUGS- BURGGERMANY. Email address:marcelorralves@gmail.com MATTHIASMEIWES. Email address:matthias.meiwes@live.de

work page 2018

[1] [1]

Alves, L

M.R.R. Alves, L. Dahinden, M. Meiwes, and L. Merlin.C 0-robustness of topological entropy for geodesic flows.J. Fixed Point Theory Appl., 24(2):42, 2022

work page 2022

[2] [2]

Alves, L

M.R.R. Alves, L. Dahinden, M. Meiwes, and A. Pirnapasov.C 0-stability of topolog- ical entropy for Reeb flows in dimension3. Published inJournal of the European Mathematical Society: online first

work page

[3] [3]

Alves and M

M.R.R. Alves and M. Meiwes. Braid stability and the Hofer metric.Annales Henri Lebesgue, 7:521–581, 2024

work page 2024

[4] [4]

Buhovsky, V

L. Buhovsky, V . Humili`ere, and S. Seyfaddini. The action spectrum andC 0 symplec- tic topology.Math. Ann., 380(1-2):293–316, 2021

work page 2021

[5] [5]

Chaidez and M

J. Chaidez and M. Hutchings. Computing Reeb dynamics on four-dimensional con- vex polytopes.J. Comput. Dyn., 8(4):403–445, 2021

work page 2021

[6] [6]

C ¸ ineli, V

E. C ¸ ineli, V . L. Ginzburg, B. Z. G¨urel, and M. Mazzucchelli. On the barcode entropy of Reeb flows.Selecta Math. (New Ser.), 2025. To appear

work page 2025

[7] [7]

Colin, P

V . Colin, P. Dehornoy, U. Hryniewicz, and A. Rechtman. Generic properties of 3-dimensional Reeb flows: Birkhoff sections and entropy.Comment. Math. Helv., 99(3):557–611, 2024

work page 2024

[8] [8]

Geiges.An introduction to contact topology, volume 109 ofCambridge Studies in Advanced Mathematics

H. Geiges.An introduction to contact topology, volume 109 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2008

work page 2008

[9] [9]

Ginzburg, B.Z

V .L. Ginzburg, B.Z. Gurel, and M. Mazzucchelli. Barcode entropy of geodesic flows. Published in Journal of the European Mathematical Society: online first., 2024

work page 2024

[10] [10]

Haim-Kislev

P. Haim-Kislev. On the symplectic size of convex polytopes.Geom. Funct. Anal., 29(2):440–463, 2019

work page 2019

[11] [11]

Haim-Kislev, R

P. Haim-Kislev, R. Hind, and Y . Ostrover. On the existence of symplectic barriers. Selecta Math. (N.S.), 30(4):65, 2024

work page 2024

[12] [12]

Haim-Kislev and Y

P. Haim-Kislev and Y . Ostrover. A counterexample to Viterbo’s conjecture.Ann. of Math. (2). To appear

work page

[13] [13]

Humili`ere, R

V . Humili`ere, R. Leclercq, and S. Seyfaddini. Coisotropic rigidity andc 0-symplectic geometry.Duke Math. J., 164(4):767–799, 2015

work page 2015

[14] [14]

A. Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes ´Etudes Sci. Publ. Math., (51):137–173, 1980. 10 MARCELO R.R. ALVES AND MATTHIAS MEIWES

work page 1980

[15] [15]

Katok and B

A. Katok and B. Hasselblatt.Introduction to the modern theory of dynamical systems, volume 54 ofEncyclopedia of Mathematics and its Applications. Cambridge Univer- sity Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza

work page 1995

[16] [16]

M. Meiwes. Topological entropy and orbit growth in link complements.Math. Ann., 393(1):617–665, 2025

work page 2025

[17] [17]

Z. Nitecki. On semi-stability for diffeomorphisms.Invent. Math., 14:83–122, 1971

work page 1971

[18] [18]

Opshtein.C 0-rigidity of characteristics in symplectic geometry.Ann

E. Opshtein.C 0-rigidity of characteristics in symplectic geometry.Ann. Sci. ´Ec. Norm. Sup´er. (4), 42(5):857–864, 2009

work page 2009

[19] [19]

Paternain.Geodesic flows, volume 180 ofProgress in Mathematics

G.P. Paternain.Geodesic flows, volume 180 ofProgress in Mathematics. Birkh ¨auser Boston, Inc., Boston, MA, 1999

work page 1999

[20] [20]

R. T. Rockafellar. Convex analysis:(pms-28). 2015

work page 2015

[21] [21]

Tsodikovich

D. Tsodikovich. Hamiltonian dynamics on simplexes. Master’s thesis, Tel Aviv Uni- versity, 2018. MARCELOR.R. ALVES, INSTITUTE OFMATHEMATICS,, UNIVERSITY OFAUGSBURG, CHAIRANALYSIS ANDGEOMETRY, UNIVERSIT ¨ATSSTRASSE14, DE-86159 AUGS- BURGGERMANY. Email address:marcelorralves@gmail.com MATTHIASMEIWES. Email address:matthias.meiwes@live.de

work page 2018