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REVIEW 3 major objections 4 minor 29 references

Scaling of the rotation number for perturbations of rational rotations

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Pith's one-line read For a circle map perturbed from a rational rotation, the rotation number is differentiable at the critical parameter when a transversality condition holds, with a Fourier-series formula for the slope.

desk verdict Explicit scaling formula for rotation number at rational rigid rotations is a real extension of Parkhe, but the proof as written has a regularity gap in Proposition 4.2 that needs patching. read the letter →

arxiv 2506.19508 v2 pith:3AKJJNRE submitted 2025-06-24 math.DS

Paul Glendinning This is my paper · ORCID
classification math.DS MSC 37E4539A28
keywords rotationnumbercirclemapsrationaldifferentiabilitymodelockingFourierseriesArnoldmappiecewiselinear
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

This paper establishes a precise condition under which the rotation number---the average speed of an orientation-preserving circle map---varies linearly with a parameter, even when the unperturbed map is a rational rotation $x \mapsto x + p/q$. The condition is a transversality inequality: the function $a+\Psi(x)$, built from the Fourier coefficients of the perturbation whose frequencies are multiples of $q$, must stay strictly positive. When it holds, $\rho(\mu)$ is differentiable at $\mu=0$ and $\rho'(0)=(\int_0^1 dx/(a+\Psi(x)))^{-1}$. If $a+\Psi$ changes sign, the rotation number is constant on a neighborhood of $\mu=0$, a mode-locked plateau with zero derivative. This contrasts with the square-root scaling found at the ends of Arnold tongues and extends classical differentiability results from irrational to rational rigid rotations.

What carries the argument

The $q$-th iterate reduction and the Euler passage-time comparison. For a perturbation of the rational rotation $p/q$, iterating the lift $q$ times yields $F^q(x,\mu)=x+p+q\mu(a+\Psi(x))+\mu^2\hat{g}(x,\mu)$, where $\Psi$ keeps exactly the Fourier coefficients of $\psi$ whose indices are multiples of $q$; translation by $p$ reduces the problem to rotation number zero. The discrete map is then read as Euler's method for the ordinary differential equation $dX/dt=a+\Psi(X)$, and the time $T_0$ the exact solution takes to cross one full period is $T_0=\int_0^1 dx/(a+\Psi(x))$. Bounding the global Euler error by $O(\mu)$ gives a rotation-number error $O(\mu^2)$, so $\rho(\mu)=T_0^{-1}\mu+O(\mu^2)$ and hence differentiability. The transversality condition $\min(a+\Psi)>0$ is precisely what keeps the ODE well-posed and the passage time finite.

What would settle it

Take the Arnold map family with $(\alpha(\mu),\beta(\mu))=(\mu,b\mu)$, so $q=1$, $a=1$, and $\psi(x)=b\sin(2\pi x)$; Theorem 1.3 predicts $\rho'(0)=1/\sqrt{1-b^2}$ for $|b|<1$. Compute $\rho(\mu)$ numerically at $\mu$ down to $10^{-5}$ and compare the measured slope with this value; any systematic deviation beyond the numerical error would refute the formula.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.3. For a family of lifts $F(x,\mu)=x+p/q+\mu(a+\psi(x))+\mu^2 g(x,\mu)$ with $\psi$ a $C^1$ mean-zero periodic function and $g$ continuous in $\mu$ and $C^1$ in $x$, define $\Psi(x)=\sum_{n\in\mathbb{Z}\setminus\{0\}} c_{nq} e^{2\pi i n q x}$ using the Fourier coefficients $c_m$ of $\psi$. If $\min_x(a+\Psi(x))>0$, then $\rho(\mu)$ is differentiable at $\mu=0$ and $\rho'(0)=T_0^{-1}$ with $T_0=\int_0^1 dx/(a+\Psi(x))$. If $\min(a+\Psi)<0<\max(a+\Psi)$, then $\rho(\mu)=p/q$ for all small $\mu$, so $\rho'(0)=0$; if $\max(a+\Psi)<0$, the same formula holds after reversing the sign of $\mu$. The mechanism is that iterating the lift $q$ times removes all non-resonant Fourier modes, so $F^q(x,\mu)=x+p+q\mu(a+\Psi(x))+O(\mu^2)$, and then the rotation number is controlled by the passage time of the differential equation $\dot{X}=a+\Psi(X)$.

Load-bearing premise

The chain of estimates needs $a+\psi(x)+\mu g(x,\mu)$ to stay bounded away from zero and to have a uniform $C^1$-in-$x$ bound with finite Lipschitz constant and a uniform bound on $|g|$ over a whole neighborhood of $\mu=0$; the paper's Definition 1.1 only assumes continuity in $\mu$ and $C^1$-in-$x$ for each fixed $\mu$, so without that extra uniformity the $O(\mu^2)$ error estimates can fail.

Editorial extensions

If this is right

  • If the theorem is correct, the rotation number near a rational rigid rotation obeys a linear scaling law whose slope is the reciprocal of the resonant passage time, so linear-response measurements can extract the resonant Fourier content of the perturbation.
  • In the Arnold map with $q\ge2$, the slope is exactly $\alpha'(0)$, independent of $\beta'(0)$, because the sine perturbation has no Fourier component of order $nq$.
  • At the origin of the Arnold map ($p/q=0/1$), the slope is $\mathrm{sgn}(\alpha'(0))\sqrt{|\alpha'(0)|^2-|\beta'(0)|^2}$ whenever $|\alpha'(0)|>|\beta'(0)|$.
  • When $a+\Psi$ changes sign, the map is mode-locked at $p/q$ on a whole neighborhood of $\mu=0$; differentiability still holds but with zero derivative.
  • The same Euler passage-time method reproduces the derivative formula for piecewise linear circle maps, giving a unified treatment of the scaling at rational rotations.

Reading between the lines

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

  • Because only the Fourier modes whose indices are multiples of $q$ enter the slope, the linear scaling acts as a spectral filter: non-resonant harmonics affect $\rho(\mu)$ only at order $\mu^2$ or higher, so the linear coefficient is a direct measurement of the resonant projection of the perturbation.
  • If the first nonzero resonant term appears only at order $\mu^k$ rather than $\mu$, the linear derivative should vanish and the leading scaling should be set by that higher-order resonant term; this is testable by adding a term $\mu^2 h(x)$ with nonzero $c_{nq}$ to the family and measuring a presumably quadratic scaling.
  • The passage-time interpretation suggests that the next coefficient in the expansion of $\rho(\mu)$ can be obtained by carrying the Euler comparison one order further, tracking the $O(\mu^2)$ term $g(x,\mu)$; such an expansion would predict the curvature of the rotation-number curve as it approaches the mode-locked plateau.
  • This result may sharpen the practical distinction between rational and irrational frequency locking: at irrational rigid rotations all harmonics contribute to the derivative, while at rational rotations only resonant harmonics do, so the slope of the rotation number carries information about the commensurability of the perturbation.
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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

3 major / 4 minor

Summary. This paper studies one-parameter families of circle homeomorphisms of the form F(x,μ)=x+p/q+μ(a+ψ(x))+μ^2 g(x,μ), where ψ is a mean-zero C^1 periodic function and g is continuous in μ and C^1 in x for each fixed μ. The main result, Theorem 1.3, states that if the Fourier-filtered function Ψ(x) built from the coefficients c_{nq} of ψ satisfies min(a+Ψ)>0, then the rotation number is differentiable at μ=0 with derivative (∫_0^1 dx/(a+Ψ(x)))^{-1}; if a+Ψ changes sign, the rotation number is locally locked at p/q; if max(a+Ψ)<0, the same formula holds after reversing the sign of μ. The proof reduces the rational case to q=1 by taking the qth iterate (Proposition 2.1), then compares the resulting difference equation with Euler steps of the ODE dX/dt=a+ψ(X)+μg(X,μ) (Proposition 4.2). The result is applied to Arnold maps, modified Arnold maps with higher harmonics, and piecewise linear circle maps, with numerical verification in each case.

Significance. If correct, Theorem 1.3 is a valuable and explicit complement to the Brunovsky-Herman results: it gives a computable derivative formula, with no fitted constants, for the rotation number at rational rigid rotations, and it explains why the local scaling is linear there rather than square-root-like at tongue boundaries. The reduction to q=1 in Proposition 2.1 is clean, the formula is testable, and the numerical comparisons are honest in the sense that they compare predicted slopes to simulations rather than tuning parameters. The result overlaps with Parkhe's theorem but offers a Fourier-series formula that is significantly easier to use. However, the proof as written has a regularity gap in the central passage-time argument and a self-referential constant in the key bound, so the main theorem is currently established only under an implicit stronger hypothesis. The paper is likely correct after a moderate revision that makes the regularity assumptions explicit and repairs the error estimates.

major comments (3)
  1. [§4, Proposition 4.2 and Definition 1.1] The proof of Proposition 4.2 requires uniform-in-μ control of g that Definition 1.1 does not provide. The Euler error bound (4.19) needs finite L = max|V'(x)| and M = max|V(x)V'(x)| on the whole μ-neighborhood, and the passage-time argument needs a uniform bound B with |g| ≤ B and the lower bound m−μ0B>0. But Definition 1.1 only assumes g is continuous in μ and C^1 in x for each fixed μ; these assumptions do not imply sup_x |g_x(x,μ)| is bounded uniformly in μ, nor even a uniform bound on |g|. For example, a periodic C^1 bump of height μ^{1/4} and width μ^{3/2} on the circle gives g→0 uniformly while g_x∼μ^{-5/4}, and μ^2g_x→0, so the map remains a small perturbation; nevertheless L and the constant k3 in (4.27) need not exist. Thus Theorem 1.3 is proved only under an implicit stronger hypothesis, such as g and g_x being bounded uniformly for |μ|<μ0. This assumption should be stated explicitly in Definition 1.1 or Proposition 4.2.
  2. [§4, Eq. (4.31)] Equation (4.31) defines k1 by k1 = T0^{-2}(20J+16+2k1), so k1 appears on both sides and the equation has no finite solution unless the constant term is zero. The preceding estimates give the fourth term in (4.29) as 2k2 μ^2/T0^2, so (4.31) should presumably read k1 = T0^{-2}(20J+16+2k2), which is explicit. As printed, the proof of Proposition 4.2 is incomplete at this point and needs a corrected, non-circular constant.
  3. [§4, Corollary 4.4 (negative μ)] The extension to μ<0 is deferred to "the same argument as in [14]". This is load-bearing because Theorem 1.3 asserts differentiability at μ=0, which requires both one-sided limits, and [14] is a preprint used as a black box here. The paper does provide an alternative proof of Theorem 7.1 for piecewise linear maps, but that proof does not by itself establish the smooth negative-μ estimate needed in Corollary 4.4. Please either prove the negative-μ case within this paper or state precisely which hypotheses from [14] are imported.
minor comments (4)
  1. [§2, Lemma 2.2] The inequality "a − ψ(x) > m1 > 0" appears to be a typo; it should presumably read "a + ψ(x) > m1 > 0" to imply F(x,μ)>x+p/q+m1μ+Bμ^2.
  2. [§4, Eq. (4.24)] The constant k2 = min{m^{-2}, 2m^{-1}M^{-1}} does not follow from the displayed estimate for |T(μ)−T0|; the natural bound is of order m^{-2}, and the role of M and the choice of min rather than max are unclear. Please check and correct.
  3. [§6, Table 1] The row (a,b,c)=(5,4,0.5) reports a numerical derivative 5.007 against a theoretical value 4.975; the text explains that a smaller μ-interval gives 4.977, which is plausible, but including error bars or a short convergence study would make the comparison more convincing.
  4. [Throughout] There are several typographical issues: "Deparment", "sh ow", a stray "S" after (4.26), "ec. 5" in §7, and the notation "10 5 iterations" for 10^5 iterations. These should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the derivative formula is derived from Fourier data and checked against simulations; self-citations are ancillary.

full rationale

The central claim, Theorem 1.3, is not circular. Proposition 2.1 computes the q-th iterate F^q exactly from the assumed lift (1.4) and the absolutely convergent Fourier series of psi; the resonant sum Psi in (2.13) is forced by the identity sum_{r=0}^{q-1} exp(2 pi i m r p/q) = q for m in qZ and zero otherwise, so it is a derivation, not a definition chosen to match (1.9). Proposition 4.2 then derives the bound (4.21) by comparing the Euler iteration (4.20) with the ODE (4.22), using the standard global error bound (4.19), the elementary integral estimate (4.24), and Lemma 4.1; T0 is computed from a+Psi, and no fitted constant or a priori value of rho'(0) is used. The numerical experiments (Figure 1b, Table 1) test the formula against independently computed rotation-number slopes rather than calibrating it. The self-citations to [14] are not load-bearing for the main theorem: Section 7 explicitly offers an alternative proof of Theorem 7.1, and Corollary 4.4's 'same argument as in [14]' is immediately followed by the actual inverse-map argument. The printed self-reference in (4.31), k1 = T0^{-2}(20J+16+2k1), is a non-circular proof defect (the preceding four-term bounds give 20J+16+2k2, so it is evidently a typo for k2), and Definition 1.1's lack of uniform C^1 control on g is a rigor gap, not a circular reduction. No prediction in the paper reduces by construction to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof draws on standard ODE and Fourier background, plus two unstated uniformity assumptions: bounded C^1-in-x control on the perturbation across the parameter neighborhood, and the homeomorphism property for small mu. No constants are fitted to data; the a, b, c, u in examples are illustrative inputs, not parameters tuned to make predictions match.

assumptions (5)
  • standard math Euler method global error bound (4.19): |X(nh)-x_n| <= k0 h with k0 = M/(2L)(e^{nhL}-1) for dX/dt=V(X) with V C^1.
    Invoked in the proof of Proposition 4.2 to bound the difference between the discrete circle map and the ODE passage time; cited to Braun [5].
  • standard math For a C^1 periodic function, its Fourier series converges absolutely, allowing interchange of summation in Proposition 2.1.
    Used in Proposition 2.1 to pass from psi(x+rp/q) to the resonant sum Psi(x); cited to Katznelson [18].
  • standard math Picard-Lindelof existence and uniqueness for the ODE dX/dt = a+psi(X)+mu g(X,mu) on [0,1] for small mu.
    Used in Proposition 4.2 to define the passage time T(mu) via an integral; cited to Braun [5].
  • domain assumption Uniform C^1 regularity and boundedness of psi and g, and positivity of a+psi(x)+mu g(x,mu), across the mu-neighborhood, so that the Euler error constants L, M, B are finite and nonzero.
    Not stated explicitly in Definition 1.1; required for the error bounds in Proposition 4.2. Pointwise continuity in mu and C^1 in x for each mu does not guarantee uniform bounds on g_x.
  • domain assumption The family F(.,mu) is a lift of an orientation-preserving circle homeomorphism for |mu|<mu0, so the rotation number is well-defined and Lemma 4.1 and monotonicity arguments apply.
    Used throughout; implied by the 'families of lifts' context and Herman's theorem but not stated as a hypothesis of Theorem 1.3.

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Pith. "Pith review of Scaling of the rotation number for perturbations of rational rotations." pith.science (2026). https://pith.science/paper/3AKJJNRE

@misc{pith2026250619508,
  author       = {Pith},
  title        = {Pith review of: Scaling of the rotation number for perturbations of rational rotations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AKJJNRE}},
  note         = {Machine review of arXiv:2506.19508}
}
abstract

The parameter dependence of the rotation number in families of circle maps which are perturbations of rational rotations is described. We show that if, at a critical parameter value, the map is a (rigid) rotation $x\to x+\frac{p}{q}~({\rm mod}~1)$ with $p$ and $q$ coprime, then the rotation number is differentiable at that point provided a transversality condition holds, and hence that the rotation number scales linearly at this parameter. We provide an explicit and computable expression for the derivative in terms of the Fourier series of the map, and illustrate the results with the Arnold circle map and some modifications. Piecewise linear circle maps can also be treated using the same techniques.

Figures

Figures reproduced from arXiv: 2506.19508 by the authors.

Figure 1
Figure 1. Rotation number ρ of (1.1) against parameter µ. Rotation numbers were computed using 105 iterations computed in Scilab [28]. In both cases 200 data points are computed at equally spaced values of µ. (a) (α(µ), β(µ)) = (0.6615+2µ, 0.1+µ), 0 < µ < 0.0003, showing the typical non-differentiable transition at the end of an Arnold tongue. (b) Two curves superposed: (α(µ), β(µ)) = ( 2 3 + 2µ, µ) and (α(µ), β(µ)) = ( 2 3 +… view at source ↗

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