Renormalization and scaling of bubbles

Igors Gorbovickis; Nataliya Goncharuk

arxiv: 2312.11308 · v2 · pith:OTLLOWKWnew · submitted 2023-12-18 · 🧮 math.DS

Renormalization and scaling of bubbles

Nataliya Goncharuk , Igors Gorbovickis This is my paper

Pith reviewed 2026-05-24 05:03 UTC · model grok-4.3

classification 🧮 math.DS

keywords renormalizationcircle diffeomorphismsbubblesscaling propertiesArnold tonguesirrational rotationscomplex dynamics

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The pith

The size of a p/q-bubble near a bounded-type irrational α is of order d^{ξ(α)} q^{-2} with ξ(α) positive.

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

The paper improves previous bounds on the size of bubbles, which are complex analogues of Arnold tongues for families of analytic circle diffeomorphisms. Earlier results established that p/q-bubbles have size at most order q to the minus two. The new work shows that when p/q lies near a bounded-type irrational α the size improves to order d to the power ξ(α) times q to the minus two, where d is the distance from α to p/q and ξ(α) is positive. The proof relies on renormalization of the circle maps, with ξ(α) coming from the ratio of eigenvalues of the renormalization operator at the irrational rotation.

Core claim

For a one-parameter family of analytic circle diffeomorphisms, the size of the p/q-bubble near a bounded-type irrational α satisfies an improved upper bound of order d^{ξ(α)} · q^{-2}, where ξ(α) > 0 is determined by the unstable and top stable eigenvalues of the renormalization operator at the rotation by α.

What carries the argument

The renormalization operator on the space of analytic circle diffeomorphisms, with its unstable and top stable eigenvalues at the fixed point corresponding to rotation by α, which determine the exponent ξ(α) in the bubble scaling.

If this is right

The bound on bubble size improves by a positive power of the distance d to the nearest bounded-type irrational.
ξ(α) is explicitly tied to the spectrum of the renormalization operator.
The result applies to all bounded-type irrationals and the given family of maps.
The scaling is stricter than the general q^{-2} bound away from such irrationals.

Where Pith is reading between the lines

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

This scaling suggests that the distribution of bubbles in the complex plane has a self-similar structure governed by renormalization at irrationals.
Numerical computation of eigenvalues for specific maps could predict bubble sizes without direct simulation.
The approach may apply to other complex extensions of rotation numbers in dynamical systems.

Load-bearing premise

The renormalization operator at the rotation by α has well-defined unstable and top stable eigenvalues that can be combined to produce a positive ξ(α) controlling the scaling.

What would settle it

For a concrete bounded-type α such as the golden mean, compute ξ(α) from the renormalization operator and then measure the actual size of bubbles for p/q with small d to check if it matches d to that power times q to the minus two.

Figures

Figures reproduced from arXiv: 2312.11308 by Igors Gorbovickis, Nataliya Goncharuk.

**Figure 1.** Figure 1: Suitable (left) and anti-suitable (right) curves for the map f with rotation number 0.5. The points a1, a3 and a2, a4 are lifts to R of the attracting and repelling periodic orbits of f of period 2. linearizing chart, the circle itself is not necessarily completely contained there. • γ passes above repelling periodic points and below attracting periodic points of f on the real line, • F(γ) is above γ in C.… view at source ↗

**Figure 2.** Figure 2: The strip A, the domain R (shadowed), and the curves ξ1, ξ2 (both shown in thick). Here rot f = 3/5, pm/qm = 1/2, and pm+1/qm+1 = 2/3. Construction of ξ1, ξ2. Assume m is even, so F qm(0) − pm > 0. Let aj ∈ R be the lifts of periodic points of f to R, ordered from left to right. Since qm < q, the interval [0, Fqm(0)−pm] contains periodic points of f; let a0 be the leftmost of these points. Take a point y ∈… view at source ↗

read the original abstract

The paper explores scaling properties of bubbles -- a complex analogue of Arnold tongues, associated to a one-dimensional family of analytic circle diffeomorphisms. Bubbles are smooth loops in the upper half-plane attached at all rational points of the real line. Results of a paper by X.~Buff and N.~Goncharuk (2015) show that the size of a $p/q$-bubble has order at most $q^{-2}$. In the current paper we improve this estimate by showing that the size of a $p/q$-bubble near a bounded-type irrational number $\alpha$ has order $d^{\xi(\alpha)} \cdot q^{-2}$, where $\xi(\alpha)>0$, and $d$ is the distance between $\alpha$ and $p/q$. Proofs are based on a renormalization technique. In particular, $\xi(\alpha)$ is related to the unstable and the top stable eigenvalues of the renormalization operator at the rotation by $\alpha$.

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

Refines the bubble-size bound near bounded-type irrationals by pulling a positive exponent from the renormalization spectrum, but the spectral gap needs explicit verification.

read the letter

The main takeaway is that the paper improves the 2015 Buff-Goncharuk bound on p/q-bubble sizes from O(q^{-2}) to O(d^ξ(α) q^{-2}) near bounded-type irrationals α, where ξ(α) > 0 is built from the unstable and top stable eigenvalues of the renormalization operator at the rotation by α. This is the concrete new piece: an explicit distance factor extracted from the same operator already used in the literature. The work stays inside the renormalization framework for analytic circle diffeomorphisms and makes the dependence on proximity to α quantitative rather than uniform. That is a modest but direct step forward for people tracking scaling in these families. The renormalization approach is the natural one here, and linking the exponent to the spectrum is a reasonable move if the operator's linearization behaves as claimed. The soft spot sits in exactly that link. The improvement requires the renormalization operator to be hyperbolic in the analytic category with the unstable eigenvalue strictly dominating the top stable one at every bounded-type rotation in the family. Standard results give hyperbolicity at certain fixed points, but the paper must show the spectral gap and the resulting positive ξ hold without extra assumptions or circularity for the full set of α considered. The abstract states the relation as given, so the manuscript needs to supply the independent spectral control and confirm that ξ is not fitted to the bubble data. If those steps are clean, the claim holds; if they rest on unstated extensions of existing hyperbolicity theorems, the improvement is weaker than presented. This is for specialists already working on renormalization of circle maps and the 2015 result. A reader who knows the operator and the prior bound will see the value in the extra factor. It is not a major reorganization, but the quantitative sharpening is worth checking. I would bring it to a reading group focused on one-dimensional dynamics. I would not cite it in my own work in the next year unless the spectral details are solid. It deserves peer review because the claim is specific, the method is standard, and the only real question is whether the eigenvalue relation is established independently.

Referee Report

2 major / 0 minor

Summary. The paper explores scaling properties of bubbles -- a complex analogue of Arnold tongues -- in a one-parameter family of analytic circle diffeomorphisms. Building on Buff-Goncharuk (2015), it claims that the size of a p/q-bubble near a bounded-type irrational α has order d^{ξ(α)} · q^{-2} with ξ(α)>0, where d is the distance from α to p/q; the exponent ξ(α) is obtained from the unstable and top stable eigenvalues of the renormalization operator at the rigid rotation by α. Proofs rely on renormalization techniques.

Significance. If the central claim holds with the required spectral gap established in the analytic category, the result would sharpen the known O(q^{-2}) bound on bubble sizes to a distance-dependent improvement controlled by renormalization eigenvalues. This would strengthen links between renormalization hyperbolicity and complex dynamics of circle maps, providing a concrete scaling law for bounded-type irrationals.

major comments (2)

[Abstract] Abstract: the claim that bubble size has order d^{ξ(α)} · q^{-2} with ξ(α)>0 is stated, but the abstract supplies no derivation steps, error estimates, or verification that the eigenvalue relation actually produces the claimed scaling; the full manuscript is required to assess whether the renormalization operator is hyperbolic with the asserted spectral properties for every bounded-type α.
[Abstract] The improved scaling rests on the existence of well-defined unstable and top stable eigenvalues whose ratio (or combination) yields positive ξ(α) controlling the bubble size; without an explicit spectral theorem or gap estimate in the analytic Banach space for the one-parameter family, it is impossible to confirm the relation is independent of the bubble-size data rather than circular.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review and comments. We address each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that bubble size has order d^{ξ(α)} · q^{-2} with ξ(α)>0 is stated, but the abstract supplies no derivation steps, error estimates, or verification that the eigenvalue relation actually produces the claimed scaling; the full manuscript is required to assess whether the renormalization operator is hyperbolic with the asserted spectral properties for every bounded-type α.

Authors: The abstract is a concise statement of the main theorem. The derivation, error estimates, and verification of hyperbolicity together with the spectral properties of the renormalization operator in the analytic Banach space (for every bounded-type α) are supplied in the body of the manuscript, where the operator is constructed and its linearization at rigid rotations is analyzed. revision: no
Referee: [Abstract] The improved scaling rests on the existence of well-defined unstable and top stable eigenvalues whose ratio (or combination) yields positive ξ(α) controlling the bubble size; without an explicit spectral theorem or gap estimate in the analytic Banach space for the one-parameter family, it is impossible to confirm the relation is independent of the bubble-size data rather than circular.

Authors: The unstable and top stable eigenvalues are computed from the linearization of the renormalization operator at the fixed point given by the rigid rotation by α. This spectral analysis is carried out in the analytic category prior to and independently of any bubble-size perturbation; the resulting positive exponent ξ(α) is therefore intrinsic to α and the scaling law is not circular. revision: no

Circularity Check

0 steps flagged

No significant circularity; scaling derived from independent spectral properties of renormalization operator

full rationale

The paper takes the q^{-2} upper bound from the 2015 Buff-Goncharuk result (self-citation but only for the baseline) and improves it to d^{ξ(α)} q^{-2} by relating bubble size to the unstable and top-stable eigenvalues of the renormalization operator at the rigid rotation by α. This is a standard dynamical-systems derivation: the operator is defined on a space of analytic circle diffeomorphisms, its linearization at the fixed point yields eigenvalues by spectral analysis, and the scaling exponent ξ(α) is extracted from their ratio or combination. No equation in the provided abstract or description defines the bubble size in terms of ξ or vice versa; the eigenvalues are properties of the operator, not fitted to the target bubble data. The derivation chain therefore remains self-contained against external benchmarks and does not reduce to a self-definition or fitted-input prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. The renormalization technique is treated as background.

pith-pipeline@v0.9.0 · 5692 in / 1069 out tokens · 21271 ms · 2026-05-24T05:03:26.158646+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/Cost/FunctionalEquation washburn_uniqueness_aczel + phi_golden_ratio echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

xi(alpha) is related to the unstable and the top stable eigenvalues of the renormalization operator at the rotation by alpha... For the golden ratio... xi = log Lambda / log phi^2... Lambda < phi^2
IndisputableMonolith/Foundation reality_from_one_distinction (hyperbolicity/renormalization aspects in wider canon) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the renormalization operator Rh : Uh -> Dh satisfies... uniformly hyperbolic on {R_alpha | alpha in BC}... unstable direction... stable leaf V_alpha

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

12 extracted references · 12 canonical work pages

[1]

V. I. Arnold. Geometrical Methods In The Theory Of Ordinary Differential Equations . Vol. 250. Grundlehren der mathematis- chen Wissenschaften [Fundamental Principles of Mathematical Science]. New York – Berlin: Springer-Verlag, 1983. 334 pp

work page 1983
[2]

Complex rotation numbers

X. Buff and N. Goncharuk. “Complex rotation numbers”. In: Journal of modern dynamics 9 (2015), pp. 169–190

work page 2015
[3]

Complex rotation numbers: bubbles and their intersections

N. Goncharuk. “Complex rotation numbers: bubbles and their intersections”. In: Analysis and PDE 11.7 (2018), pp. 1787–1801. doi: 10.2140/apde.2018.11.1787

work page doi:10.2140/apde.2018.11.1787 2018
[4]

Self-similarity of bubbles

N. Goncharuk. “Self-similarity of bubbles”. In: Nonlinearity 32.7 (June 2019), pp. 2496–2521. doi: 10.1088/1361-6544/ab1b8f . url: https://doi.org/10.1088/1361-6544/ab1b8f

work page doi:10.1088/1361-6544/ab1b8f 2019
[5]

Analytic linearization of con- formal maps of the annulus

N. Goncharuk and M. Yampolsky. “Analytic linearization of con- formal maps of the annulus”. In: Advances in Mathematics 409 (2022), p. 108636. issn: 0001-8708. doi: https://doi.org/10. 1016/j.aim.2022.108636. url: https://www.sciencedirect. com/science/article/pii/S0001870822004534

work page arXiv 2022
[6]

Morse-Smale circle diffeomor- phisms and moduli of complex tori

Y. Ilyashenko and V. Moldavskis. “Morse-Smale circle diffeomor- phisms and moduli of complex tori”. In: Moscow Mathematical Journal 3.2 (April-June 2003), pp. 531–540

work page 2003
[7]

Rotation numbers and moduli of elliptic curves

N.Goncharuk. “Rotation numbers and moduli of elliptic curves”. In: Functional Analysis and Its Applications 46.1 (2012), pp. 11– 25

work page 2012
[8]

A note on the inclination Lemma ( λ-Lemma) and Feigen- baum’s rate of approach

J. Palis. “A note on the inclination Lemma ( λ-Lemma) and Feigen- baum’s rate of approach”. In: Palis, J. (eds) Geometric Dynam- ics. Lecture Notes in Mathematics, vol. 1007 (1983)

work page 1983
[9]

Lin´ earisation des perturbations holomorphes des ro- tations et applications

E. Risler. “Lin´ earisation des perturbations holomorphes des ro- tations et applications”. In : M´ emoires de la S.M.F. 2e s´ er. 77 (1999), p. 1-102

work page 1999
[10]

Moduli of Elliptic Curves and Rotation Numbers of Circle Diffeomorphisms

V.Moldavskij. “Moduli of Elliptic Curves and Rotation Numbers of Circle Diffeomorphisms”. In: Functional Analysis and Its Ap- plications 35.3 (2001), pp. 234–236. 30 REFERENCES

work page 2001
[11]

Inclination lemmas with dominated convergence

H.-O. Walter. “Inclination lemmas with dominated convergence”. In: Z. angew. Math. Phys. 38 (1987), pp. 327–337. doi: https: //doi.org/10.1007/BF00945417

work page doi:10.1007/bf00945417 1987
[12]

Conjugaison diff´ erentiable des diff´ eomorphismes du cercle dont le nombre de rotation v´ erifie une condition dio- phantienne

J.-C. Yoccoz. “Conjugaison diff´ erentiable des diff´ eomorphismes du cercle dont le nombre de rotation v´ erifie une condition dio- phantienne”. In : Annales Scientifiques de l’ ´Ecole Normale Su- perieure. 4e s´ er. 3 (17 1984), p. 333-359

work page 1984

[1] [1]

V. I. Arnold. Geometrical Methods In The Theory Of Ordinary Differential Equations . Vol. 250. Grundlehren der mathematis- chen Wissenschaften [Fundamental Principles of Mathematical Science]. New York – Berlin: Springer-Verlag, 1983. 334 pp

work page 1983

[2] [2]

Complex rotation numbers

X. Buff and N. Goncharuk. “Complex rotation numbers”. In: Journal of modern dynamics 9 (2015), pp. 169–190

work page 2015

[3] [3]

Complex rotation numbers: bubbles and their intersections

N. Goncharuk. “Complex rotation numbers: bubbles and their intersections”. In: Analysis and PDE 11.7 (2018), pp. 1787–1801. doi: 10.2140/apde.2018.11.1787

work page doi:10.2140/apde.2018.11.1787 2018

[4] [4]

Self-similarity of bubbles

N. Goncharuk. “Self-similarity of bubbles”. In: Nonlinearity 32.7 (June 2019), pp. 2496–2521. doi: 10.1088/1361-6544/ab1b8f . url: https://doi.org/10.1088/1361-6544/ab1b8f

work page doi:10.1088/1361-6544/ab1b8f 2019

[5] [5]

Analytic linearization of con- formal maps of the annulus

N. Goncharuk and M. Yampolsky. “Analytic linearization of con- formal maps of the annulus”. In: Advances in Mathematics 409 (2022), p. 108636. issn: 0001-8708. doi: https://doi.org/10. 1016/j.aim.2022.108636. url: https://www.sciencedirect. com/science/article/pii/S0001870822004534

work page arXiv 2022

[6] [6]

Morse-Smale circle diffeomor- phisms and moduli of complex tori

Y. Ilyashenko and V. Moldavskis. “Morse-Smale circle diffeomor- phisms and moduli of complex tori”. In: Moscow Mathematical Journal 3.2 (April-June 2003), pp. 531–540

work page 2003

[7] [7]

Rotation numbers and moduli of elliptic curves

N.Goncharuk. “Rotation numbers and moduli of elliptic curves”. In: Functional Analysis and Its Applications 46.1 (2012), pp. 11– 25

work page 2012

[8] [8]

A note on the inclination Lemma ( λ-Lemma) and Feigen- baum’s rate of approach

J. Palis. “A note on the inclination Lemma ( λ-Lemma) and Feigen- baum’s rate of approach”. In: Palis, J. (eds) Geometric Dynam- ics. Lecture Notes in Mathematics, vol. 1007 (1983)

work page 1983

[9] [9]

Lin´ earisation des perturbations holomorphes des ro- tations et applications

E. Risler. “Lin´ earisation des perturbations holomorphes des ro- tations et applications”. In : M´ emoires de la S.M.F. 2e s´ er. 77 (1999), p. 1-102

work page 1999

[10] [10]

Moduli of Elliptic Curves and Rotation Numbers of Circle Diffeomorphisms

V.Moldavskij. “Moduli of Elliptic Curves and Rotation Numbers of Circle Diffeomorphisms”. In: Functional Analysis and Its Ap- plications 35.3 (2001), pp. 234–236. 30 REFERENCES

work page 2001

[11] [11]

Inclination lemmas with dominated convergence

H.-O. Walter. “Inclination lemmas with dominated convergence”. In: Z. angew. Math. Phys. 38 (1987), pp. 327–337. doi: https: //doi.org/10.1007/BF00945417

work page doi:10.1007/bf00945417 1987

[12] [12]

Conjugaison diff´ erentiable des diff´ eomorphismes du cercle dont le nombre de rotation v´ erifie une condition dio- phantienne

J.-C. Yoccoz. “Conjugaison diff´ erentiable des diff´ eomorphismes du cercle dont le nombre de rotation v´ erifie une condition dio- phantienne”. In : Annales Scientifiques de l’ ´Ecole Normale Su- perieure. 4e s´ er. 3 (17 1984), p. 333-359

work page 1984