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On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in $\mathbb{R}^{2n}$

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Pith's one-line read On every dynamically convex star-shaped hypersurface in $\mathbb{R}^{2n}$ with finitely many closed characteristics, at least $\lfloor(n+1)/2\rfloor$ have irrational mean index and at least $\lfloor(n+1)/2\rfloor+1$ have pairwise…

desk verdict Real results for n≥2, but the main theorems are false as stated because they include n=1. read the letter →

arxiv 2506.04546 v1 pith:4P2BXJN4 submitted 2025-06-05 math.SG

Wei Wang This is my paper
classification math.SG MSC 58E0537J4634C25
keywords closedcharacteristicsstar-shapedhypersurfacesdynamicallyconvexmeanindexequivariantsymplectichomologyLusternik-SchnirelmanntheoryReeborbitsirrationalindices
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 any dynamically convex compact star-shaped hypersurface in $\mathbb{R}^{2n}$ with only finitely many closed characteristics, the paper proves that at least $\lfloor(n+1)/2\rfloor$ geometrically distinct closed characteristics have irrational mean index, and that at least $\lfloor(n+1)/2\rfloor+1$ can be chosen so that the ratio of the mean indices of any two is irrational. This improves the previously known convex-hypersurface bounds by one orbit whenever $n$ is odd, and the same conclusions follow for strictly convex hypersurfaces as corollaries. The result matters because guaranteed irrational mean indices are a sharp way to express that periodic orbits are not all of the same rational-index type; the proof shows the extra orbit comes from the structure of equivariant symplectic homology rather than from convexity.

What carries the argument

The load-bearing object is the Lusternik-Schnirelmann theory for the shift operator $D$ in positive equivariant symplectic homology $SH^{S^1,+}_*(W;\mathbb{Q})$. The shift operator gives a strictly increasing sequence of spectral invariants and a carrier map that assigns to each degree $d$ an orbit $y_d$ with nonzero local equivariant symplectic homology; the index recurrence theorem, with the integer sequences $d_j$ and $k_{ij}$ chosen by the common index jump construction, makes the window $\tilde{L}=[d_j-2,d_j+n]\cap\{n+1,n+3,\dots\}$ consist of distinct prime-orbit iterations. The resonance relation $A_\alpha(x)/\hat{\mu}(x)=\text{const}$ for reoccurring orbits then converts any rationality among mean indices in the window into equality of actions, which contradicts the strict inequality $A_\alpha(y_{d_1})\neq A_\alpha(y_{d_2})$ for $d_1\neq d_2$.

What would settle it

Compute the mean indices and the equivariant local symplectic homology ranks in the window $[d_j-2,d_j+n]$ for an explicit dynamically convex star-shaped hypersurface with finitely many closed orbits, such as an irrational ellipsoid; if a degree in that window is supported by a non-reoccurring orbit, or if fewer than $\lfloor(n+1)/2\rfloor$ of the window's supporting orbits have irrational mean index, the theorem's resonance-based contradiction fails.

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Extended reading notes

Core claim

The central discovery is that dynamical convexity plus finiteness forces irrational mean indices in a quantitative way: among the finitely many geometrically distinct prime closed characteristics of $\Sigma$, at least $\lfloor(n+1)/2\rfloor$ have $\hat{\mu}(x)\notin\mathbb{Q}$, and at least $\lfloor(n+1)/2\rfloor+1$ have pairwise irrational mean-index ratios. The proof works by placing an entire window of degrees $[d_j-2,d_j+n]$ inside the image of the carrier map coming from equivariant symplectic homology, with each degree occupied by a different prime orbit's iteration. The extra degree $d_j-2$ is available precisely because a symplectically degenerate maximum would generate infinitely many closed characteristics, contradicting finiteness. Once two such orbits were to have rational-related mean indices, the resonance relation would force their actions to coincide, contradicting the strict monotonicity of spectral invariants.

Load-bearing premise

The load-bearing premise is that the labeling of homology degrees by orbits (the carrier map) can be chosen so that every orbit in the index-jump window is reoccurring, meaning its iterations appear infinitely often, because only then does the resonance relation apply and produce the action contradiction that both theorems use; the paper invokes a known theorem for this choice rather than constructing it.

Editorial extensions

If this is right

  • For odd $n$, the previous convex-hypersurface bound of $\lfloor n/2\rfloor$ irrational-mean-index orbits is superseded by $\lfloor(n+1)/2\rfloor$, one more orbit.
  • The family of orbits with pairwise irrational mean-index ratios has size $\lfloor(n+1)/2\rfloor+1$, one larger than the earlier convex bound when $n$ is odd.
  • Both theorems hold for all dynamically convex star-shaped hypersurfaces, not only strictly convex ones, so the corollaries extend the convex results to a wider class of energy surfaces.
  • The bounds are sharp at $n=3$: the harmonic-oscillator example has three prime orbits, with two irrational mean indices and all three pairwise ratios irrational, so no stronger uniform bound is possible in that dimension.
  • The finiteness hypothesis is essential: the equal-frequency harmonic oscillator has infinitely many closed characteristics and none with irrational mean index, so the conclusion can fail without it.

Reading between the lines

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

  • The same index-window mechanism should transfer to any contact manifold whose positive equivariant symplectic homology carries a shift operator, spectral invariants, and a resonance relation for reoccurring orbits; the paper works in $\mathbb{R}^{2n}$, but the mechanism is not obviously tied to that embedding.
  • Since hyperbolic orbits have rational mean index under the standard conventions, Theorem 1.1 would in particular guarantee $\lfloor(n+1)/2\rfloor$ non-hyperbolic closed characteristics; the paper does not draw this stability conclusion explicitly.
  • The dimension parameter should be read as $n\ge2$: for $n=1$, a convex planar energy curve has a single closed characteristic with rational mean index, so the lower bound as written would not hold without that convention.
  • Whether the odd-dimensional gain of one orbit persists in higher even dimensions is left open; testing the bounds on irrational ellipsoids in $\mathbb{R}^8$ and $\mathbb{R}^{10}$ would show how close the general estimates are to the actual minimal counts.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies closed characteristics on compact star-shaped hypersurfaces in R^{2n} that are dynamically convex and have only finitely many geometrically distinct closed characteristics. Theorem 1.1 claims that at least floor((n+1)/2) such characteristics have irrational mean index; Theorem 1.2 claims that at least floor((n+1)/2)+1 have pairwise irrational mean-index ratios; Corollaries 1.3 and 1.4 state the corresponding results for strictly convex hypersurfaces. The proofs apply the Lusternik-Schnirelmann theory for the shift operator in equivariant symplectic homology developed by Ginzburg and Gürel, together with the index recurrence theorem, resonance relations, and the symplectically degenerate maximum argument. The claimed improvement over Long-Zhu and Hu-Ou is by one orbit when n is odd, and the paper gives explicit harmonic-oscillator examples to show sharpness for n=3.

Significance. If the statements are restricted to n>=2, the paper is a meaningful contribution: it combines the [GG] Lusternik-Schnirelmann framework with Long-Zhu common-index-jump estimates, and it provides explicit, checkable examples including the sharp n=3 case in Remark 1.6. The proofs are detailed, and their dependence on external results is largely transparent. The principal weakness is that the main theorems as stated are false for n=1, which makes the significance conditional on a corrected n>=2 statement and on filling a gap in the reoccurring-orbit step.

major comments (2)
  1. [Section 1, Theorems 1.1 and 1.2 and Corollaries 1.3 and 1.4; Remark 1.6] The statements are false for n=1. Take Sigma=E_1(1), the unit circle in R^2. It is strictly convex, hence belongs to H_con(2) and is dynamically convex by Theorem 2.6. By the paper's own Remark 1.6, #T(Sigma)=1 and the unique orbit has mean index \hat i(x_1)=2, which is rational. Therefore Theorem 1.1, which would require at least floor(2/2)=1 orbit with irrational mean index, and Theorem 1.2, which would require two orbits, both fail; Corollaries 1.3 and 1.4 fail for the same example even under the weaker convexity assumption. All main statements and the proof must be restricted to n>=2, and that restriction must be explicit, because for n=1 the formalism with m=n-1 in Section 2 is not the intended setting.
  2. [Section 3, proof of Theorem 1.1 after Eq. (3.7), and the analogous step in Theorem 1.2] The sentence 'Since the number of prime closed Reeb orbits is finite, we can choose x_d to be reoccurring for all d in \tilde L, cf. Theorem 6.1 of [GG]' is not a complete justification. Proposition 2.3's resonance relation (2.10) applies only to reoccurring orbits, and the equalities (3.8)-(3.9) and (3.24) depend on that relation. The author should prove that the carrier map can be arranged so that every orbit appearing in the common index jump interval is reoccurring, for instance by taking d_j larger than the maximal degree of any non-reoccurring orbit, whose existence follows from assumption (F) and the definition of reoccurring. As written, this is a gap in a load-bearing step of both proofs.
minor comments (5)
  1. [Section 3, first line] The phrase 'In his section' should be 'In this section'.
  2. [Abstract and Theorem 1.2] The phrase 'a irrational number' should be 'an irrational number'; the same grammatical issue appears in the abstract and in Theorem 1.2.
  3. [References] The entries [Eke] and [Eke2] appear to be the same paper with the same title, year, and page range; one of them should be removed or corrected.
  4. [Theorem 2.4] The phrase 'r integer sequences k_{ij}' is ambiguous; using k_{i,j} or explicitly indicating the index ranges would improve clarity.
  5. [Section 2, page 6] The word 'differmorphic' should be 'diffeomorphic'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are derived from external index-recurrence and resonance results, not from fitted inputs or self-citations.

full rationale

The proof of Theorems 1.1 and 1.2 rests on Proposition 2.1 (carrier map, Corollary 3.9 of [GG]), Theorem 2.4 (index recurrence, Theorem 5.2 of [GG]), Proposition 2.7 (dynamical convexity bounds, Corollary 5.4 of [GG]), Proposition 2.3 (resonance relation, Theorem 6.4 of [GG]), and the SDM argument of [GHHM]. None of these are the author's own results, and none are fed with the target conclusion. The choice of the integer sequences d_j and k_ij in Theorem 2.4 is a number-theoretic construction taken from [LoZ] (explicitly, eq. (3.12) and Claim 3); it does not encode rationality or irrationality of mean indices. In the proof of Claim 2, the rationality of two mean indices is used only to force equality (3.7), and the contradiction comes from the strict monotonicity of actions in (3.3), an independent input. The author's own prior works [Wan1], [Wan3], [Wan4], and [WHL] appear only as background citations in the introduction or in Remark 1.6 as examples; the proof never reduces to them. Two correctness issues are flagged but are not circularity: (i) as written, Theorems 1.1-1.2 and Corollaries 1.3-1.4 fail for n=1 (e.g., the unit circle in R^2 by Remark 1.6 has one closed characteristic with integer mean index 2, contradicting the required counts); the statements need the hypothesis n>=2. (ii) The assertion 'we can choose x_d to be reoccurring for all d in Ltilde, cf. Theorem 6.1 of [GG]' (Section 3, after Claim 2) is invoked without a detailed proof; if Theorem 6.1 of [GG] does not cover exactly this carrier-map selection, that is a gap in justification, but it is not a reduction-by-definition of the conclusion. No fitted parameter is relabeled as a prediction and no input is defined in terms of the output.

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

The central claim rests on a stack of published theorems from equivariant symplectic homology and index theory; no free parameters or invented entities are introduced. The n=1 failure is a statement-level error, not a ledger item.

assumptions (6)
  • domain assumption Existence, monotonicity, and index bounds of the equivariant symplectic homology spectral invariants and carrier map (Proposition 2.1).
    Imported from [GG] Corollary 3.9; used to attach to each degree d a closed Reeb orbit y_d with ordered actions and bounded mean index.
  • domain assumption Index recurrence theorem with divisibility (Theorem 2.4).
    Imported from [GG] Theorem 5.2; the divisibility of d_j and k_ij by any N is used in Claim 2. A new proof is only sketched using [LoZ].
  • domain assumption Symplectically degenerate maximum forces infinitely many geometrically distinct closed Reeb orbits.
    Imported from [GHHM]; used to eliminate y_{d_j-2}=x_i^{k_ij-1} and obtain the extra orbit for odd n.
  • domain assumption Resonance relations for reoccurring closed characteristics (Proposition 2.3).
    Imported from [GG] Theorem 6.4; used in Claims 2 and 4 to convert equal mean indices into equal actions.
  • domain assumption Dynamically convex Reeb flow on strictly convex hypersurfaces (Theorem 2.6).
    Imported from [GG] Theorem 4.10; this yields Corollaries 1.3 and 1.4. It is stated without an n≥2 restriction, which is part of the n=1 problem.
  • domain assumption Long-Zhu common index jump estimates and the vertex Diophantine construction.
    Imported from [LoZ] equations (4.10), (4.40), (4.41) and Theorem 4.2; used to construct k_i explicitly in Theorem 1.2.

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Pith. "Pith review of On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in $\mathbb{R}^{2n}$." pith.science (2026). https://pith.science/paper/4P2BXJN4

@misc{pith2026250604546,
  author       = {Pith},
  title        = {Pith review of: On the mean indices of closed characteristics on dynamically convex star-shaped hypersurfaces in $\mathbbR^2n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4P2BXJN4}},
  note         = {Machine review of arXiv:2506.04546}
}
abstract

In this paper, we prove that for every dynamically convex compact star-shaped hypersurface $\Sigma\subset\mathbb{R}^{2n}$, there exist at least $\lfloor\frac{n+1}{2}\rfloor$ geometrically distinct closed characteristics possessing irrational mean indices provided the number of geometrically distinct closed characteristics on $\Sigma$ is finite, this improves Theorem 1.3 in \cite{LoZ} of Y. Long and C. Zhu by finding one more closed characteristic possessing irrational mean index when $n$ is odd. Moreover, there exist at least $\lfloor\frac{n+1}{2}\rfloor+1$ geometrically distinct closed characteristics such that the ratio of the mean indices of any two of them is a irrational number provided the number of geometrically distinct closed characteristics on $\Sigma$ is finite, this improves Theorem 1.2 in \cite{HuO} of X. Hu and Y. Ou when $n$ is odd. In particular, these estimates are sharp for $n=3$.

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