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arxiv: 2606.13372 · v1 · pith:IULHKDULnew · submitted 2026-06-11 · 🧮 math.SG · math.DS

Filtered Symplectic Homology and Closed Reeb Orbits

Erman Cineli , Viktor L. Ginzburg , Basak Z. Gurel This is my paper

Pith reviewed 2026-06-27 04:45 UTC · model grok-4.3

classification 🧮 math.SG math.DS
keywords symplectic homologyReeb orbitsmean indexstar-shaped domainspseudo-rotationsfiltered homologyS1-equivariant invariantscontact geometry
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The pith

A non-degenerate Reeb flow on a star-shaped domain has infinitely many prime closed orbits if it has one with negative mean index.

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

The paper links the filtered symplectic homology persistence module to the number and indices of closed Reeb orbits on star-shaped domains in higher dimensions. It shows that S¹-equivariant spectral invariants stay bounded above over fields of positive characteristic, unlike the zero-characteristic case. When the Reeb flow is a pseudo-rotation with only finitely many prime orbits, the dimension of the filtered homology remains bounded as a function of action. The central result establishes that the presence of one closed orbit with negative mean index forces infinitely many distinct prime closed orbits under the non-degeneracy assumption.

Core claim

If a non-degenerate Reeb flow on a star-shaped domain has a closed orbit with negative mean index, then it possesses infinitely many prime closed orbits. The proof proceeds by relating the mean index to the growth properties of the filtered symplectic homology and its S¹-equivariant spectral invariants.

What carries the argument

The filtered symplectic homology persistence module, which tracks action values and mean indices of Reeb orbits to detect whether the total number of prime orbits must be infinite.

If this is right

  • S¹-equivariant spectral invariants remain bounded above when working over a field of positive characteristic.
  • The dimension of filtered symplectic homology is bounded by a function of the action level for any pseudo-rotation.
  • Existence of one negative-mean-index orbit is incompatible with having only finitely many prime closed orbits.
  • The infinitude result applies in all dimensions where the domain is star-shaped.

Where Pith is reading between the lines

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

  • The boundedness statements may constrain the possible action spectra of flows with only finitely many orbits.
  • Similar infinitude conclusions could hold for other classes of contact manifolds if the homology persistence module behaves analogously.
  • The distinction between positive and zero characteristic suggests that algebraic properties of the coefficient field affect orbit-counting conclusions.

Load-bearing premise

The Reeb flow is non-degenerate and the domain is star-shaped.

What would settle it

A single counterexample consisting of a non-degenerate Reeb flow on a star-shaped domain that has exactly one closed orbit with negative mean index yet only finitely many prime closed orbits in total.

read the original abstract

We further explore connections between the symplectic homology persistence module and the properties of closed Reeb orbits for star-shaped domains in higher dimensions. Our first result is that the sequence of $S^1$-equivariant spectral invariants over a field of positive characteristic is bounded from above, in contrast with the case of characteristic zero. We also prove that the dimension of the filtered symplectic homology is bounded as a function of the action whenever the flow is a pseudo-rotation, i.e., it has finitely many prime closed orbits. Finally, we show that a non-degenerate Reeb flow has infinitely many prime closed orbits whenever it has one closed orbit with negative mean index.

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.

Referee Report

0 major / 2 minor

Summary. The paper establishes three results on filtered symplectic homology for star-shaped domains in higher dimensions: (i) the sequence of S¹-equivariant spectral invariants is bounded above over fields of positive characteristic (in contrast to characteristic zero); (ii) the dimension of the filtered symplectic homology is bounded as a function of the action when the Reeb flow is a pseudo-rotation (finitely many prime closed orbits); (iii) a non-degenerate Reeb flow has infinitely many prime closed orbits if it possesses at least one closed orbit of negative mean index.

Significance. If the claims hold, the work strengthens the link between persistence modules in symplectic homology and Reeb dynamics, particularly by providing a criterion for orbit infinitude based on mean index and by highlighting characteristic-dependent phenomena in spectral invariants. The pseudo-rotation bound offers a quantitative control that may be useful for studying finite-orbit cases in contact geometry.

minor comments (2)
  1. The abstract states the infinitude result for non-degenerate flows on star-shaped domains but does not indicate the precise range of dimensions (beyond 'higher dimensions'); the introduction should clarify the minimal dimension required for the constructions.
  2. Notation for the filtered symplectic homology and the S¹-equivariant spectral invariants should be introduced with explicit references to the persistence module structure used in the proofs of the three theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the paper and for recognizing the significance of the results linking persistence modules in symplectic homology to Reeb dynamics. The summary accurately captures the three main theorems. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's results on bounded S^1-equivariant spectral invariants in positive characteristic, dimension bounds for filtered symplectic homology under pseudo-rotations, and infinitude of prime closed Reeb orbits given one with negative mean index are derived from standard filtered symplectic homology constructions and non-degeneracy assumptions on star-shaped domains. No step reduces a claimed prediction or theorem to a fitted parameter, self-definition, or load-bearing self-citation chain; the derivation chain remains independent of its target outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the claims rest on standard background theory of filtered symplectic homology for star-shaped domains and properties of Reeb flows; no free parameters or new entities are visible.

axioms (2)
  • domain assumption Standard properties of S1-equivariant symplectic homology persistence modules hold for star-shaped domains in higher dimensions
    Invoked implicitly for all three results; the paper assumes the established algebraic framework from prior literature.
  • domain assumption The Reeb flow is non-degenerate when applying the infinitude result
    Explicitly stated in the final result of the abstract.

pith-pipeline@v0.9.1-grok · 5641 in / 1275 out tokens · 46528 ms · 2026-06-27T04:45:22.664342+00:00 · methodology

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Reference graph

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