arxiv: 2605.12463 · v1 · submitted 2026-05-12 · 🧮 math.SG · math.DS· math.GT

Recognition: 2 theorem links

· Lean Theorem

On the growth rate of Reeb orbit on star-shaped hypersurfaces

Joao Pering, Rafael Fernandes

Pith reviewed 2026-05-13 02:10 UTC · model grok-4.3

classification 🧮 math.SG math.DSmath.GT

keywords Reeb orbitsgrowth ratestar-shaped hypersurfacescotangent bundleclosed geodesicsChas-Sullivan productsymplectic homologyLiouville domains

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The pith

The number of simple Reeb orbits with period at most T grows at least like T over log T on star-shaped hypersurfaces in cotangent bundles under a given topological condition.

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

The paper shows that for fiberwise star-shaped hypersurfaces in the cotangent bundle of a closed manifold with a certain topological property, the Reeb flow has infinitely many simple closed orbits. Specifically, the count of those with period bounded by T is at least comparable to the number of primes up to T, which is T/log(T). This condition involves a non-nilpotent homology class in the free loop space via the Chas-Sullivan product, linked to a non-torsion class in first homology. The result applies in particular to the growth of closed geodesics on such manifolds and extends to certain Liouville domains via symplectic homology.

Core claim

Under the assumption of a non-nilpotent class in the homology of the free loop space with respect to the Chas-Sullivan product in a component associated to a non-torsion first homology class, any fiberwise star-shaped hypersurface in the cotangent bundle carries infinitely many simple Reeb orbits, with the number of period at most T growing at least like T/log(T).

What carries the argument

The non-nilpotent class in the homology of the free loop space of the manifold with respect to the Chas-Sullivan product, associated to a non-torsion class in first homology.

Load-bearing premise

The existence of a non-nilpotent class in the homology of the free loop space of the manifold with respect to the Chas-Sullivan product, lying in a connected component associated to a non-torsion class in the first homology of the manifold.

What would settle it

A manifold satisfying the topological condition but equipped with a star-shaped hypersurface whose simple Reeb orbits number o(T/log T) up to period T, or only finitely many, would disprove the claim.

Figures

Figures reproduced from arXiv: 2605.12463 by Joao Pering, Rafael Fernandes.

read the original abstract

In this article, we study the growth rate of Reeb orbits on fiberwise star-shaped hypersurfaces in the cotangent bundle of a closed manifold. We prove that under a suitable topological condition on the base manifold the Reeb flow on any such hypersurface carries infinitely many simple closed orbits. Moreover, the number of simple Reeb orbits with period at most T grows at least like the prime numbers, that is, like T/log(T). The topological condition we assume is the existence of a non-nilpotent class in the homology of the free loop space of the manifold, with respect to the Chas-Sullivan product, lying in a connected component associated to a non-torsion class in the first homology of the manifold. In particular, for any Riemannian metric on a manifold satisfying such a topological condition, the number of geometrically distinct closed geodesics with length at most l grows at least like l/log(l). We also prove, using symplectic homology, that if a Liouville domain of dimension at least 4 with vanishing first Chern class admits a Reeb symplectically degenerate maximum representing a non-torsion first homology class of the domain, then the number of simple Reeb orbits with period at most T grows at least like T/log(T).

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

This paper gives a T/log(T) lower bound on simple Reeb orbits for star-shaped hypersurfaces when the base has a non-nilpotent Chas-Sullivan class tied to non-torsion H1.

read the letter

The paper claims that if a closed manifold has a non-nilpotent class in the homology of its free loop space with respect to the Chas-Sullivan product, and that class is in a component linked to a non-torsion element in first homology, then any fiberwise star-shaped hypersurface in its cotangent bundle has at least T/log(T) simple closed Reeb orbits of period at most T. The same holds for closed geodesics on the manifold itself. There's an additional statement for certain Liouville domains using symplectic homology and a symplectically degenerate maximum representing a non-torsion class. This is new in giving a specific growth rate rather than just infinitely many orbits. It connects algebraic topology directly to dynamical counts in a way that builds on known existence results. The approach looks reasonable because it uses established tools like symplectic homology to translate the non-nilpotency into orbit growth. The condition is stated clearly, which helps. One potential issue is that the growth bound depends on how the homology class interacts with the action filtration in the symplectic homology. If the paper shows that non-nilpotency produces arbitrarily high iterates or distinct orbits in a controlled way, it works. But details on error estimates or exact constants aren't in the abstract, so the full text needs to confirm no gaps in the counting. This work is for researchers in symplectic and contact geometry focused on Reeb flows and closed geodesics. It gives a concrete quantitative result under a verifiable topological assumption. I recommend sending it for peer review. The idea is sharp enough and the methods standard enough that referees can assess the details properly.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that, under the topological hypothesis that the free loop space of a closed manifold M carries a non-nilpotent class (with respect to the Chas-Sullivan product) lying in a connected component corresponding to a non-torsion class in H_1(M), any fiberwise star-shaped hypersurface in T^*M has Reeb flow with at least T/log(T) simple closed orbits of period at most T. The same growth rate is obtained for closed geodesics on such manifolds. A parallel result is proved for Liouville domains of dimension at least 4 with vanishing first Chern class that admit a Reeb symplectically degenerate maximum representing a non-torsion class in H_1, via symplectic homology.

Significance. If the proofs are correct, the work supplies a quantitative lower bound on the number of simple Reeb orbits that is asymptotically stronger than mere infinitude and is controlled by a standard algebraic-topological invariant. The reduction of the geodesic problem to the same statement is a clean application. The use of established tools (symplectic homology, Chas-Sullivan product) without additional ad-hoc parameters is a methodological strength.

minor comments (3)

[Introduction] The abstract states that the T/log(T) bound follows from the non-nilpotency hypothesis via symplectic homology or loop-space homology, but the introduction should include a short roadmap (one paragraph) indicating which filtration or action window is used to extract the prime-number growth from the algebraic non-nilpotency.
[Main theorem for cotangent bundles] In the statement of the main theorem for star-shaped hypersurfaces, the precise relation between the period T and the action filtration on the symplectic homology should be made explicit (e.g., which multiple of the minimal action is taken).
[Liouville domain result] The definition of a 'Reeb symplectically degenerate maximum' in the Liouville-domain theorem should be recalled or referenced at the point where it is first used, together with a one-sentence explanation of why it produces a non-nilpotent class.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. We are grateful for the recognition of the significance of our quantitative lower bounds on the number of simple Reeb orbits and closed geodesics under the stated topological hypotheses.

read point-by-point responses

Referee: The manuscript proves that, under the topological hypothesis that the free loop space of a closed manifold M carries a non-nilpotent class (with respect to the Chas-Sullivan product) lying in a connected component corresponding to a non-torsion class in H_1(M), any fiberwise star-shaped hypersurface in T^*M has Reeb flow with at least T/log(T) simple closed orbits of period at most T. The same growth rate is obtained for closed geodesics on such manifolds. A parallel result is proved for Liouville domains of dimension at least 4 with vanishing first Chern class that admit a Reeb symplectically degenerate maximum representing a non-torsion class in H_1, via symplectic homology.

Authors: We thank the referee for this precise encapsulation of our main results. The description correctly reflects the non-nilpotency condition in free loop space homology, the resulting T/log(T) growth for Reeb orbits on star-shaped hypersurfaces in cotangent bundles, the reduction to closed geodesics, and the parallel statement for Liouville domains via symplectic homology. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external tools

full rationale

The paper conditions its T/log(T) lower bound on the existence of a non-nilpotent Chas-Sullivan class in a non-torsion H1-component of the free loop space, which is an external topological input rather than an internally fitted or self-defined quantity. The growth estimate is obtained by feeding this hypothesis into standard symplectic homology constructions and the action filtration on the contact homology of star-shaped hypersurfaces; no equation or step equates the output bound to a parameter fitted from the same data, nor does any load-bearing premise reduce to a self-citation whose content is unverified. The corollary for closed geodesics and the Liouville-domain result likewise invoke only external algebraic topology and symplectic homology without internal reduction. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background results in symplectic and algebraic topology without introducing new fitted parameters or postulated entities.

axioms (2)

standard math Symplectic homology is well-defined and functorial for Liouville domains with vanishing first Chern class
Invoked for the second theorem on Liouville domains
standard math The Chas-Sullivan product is defined on the homology of the free loop space and detects non-nilpotency
Central to the topological condition in the first theorem

pith-pipeline@v0.9.0 · 5523 in / 1355 out tokens · 60442 ms · 2026-05-13T02:10:09.224359+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the existence of a non-nilpotent class in the homology of the free loop space of the manifold, with respect to the Chas–Sullivan product, lying in a connected component associated to a non-torsion class in the first homology of the manifold
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_strictMono unclear
lim inf N_Σ^η(T) log(T)/T > 0

Reference graph

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