[Paper Review] Smooth Siegel disks via semicontinuity: a remark on a proof of Buff and Cheritat

Artur Avila|ArXiv.org|May 19, 2003

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper provides a streamlined proof of the existence of quadratic Siegel disks with smooth (C∞) boundaries using semicontinuity arguments and known results from Yoccoz and Risler on renormalization theory. By leveraging upper and weak lower semicontinuity of the conformal radius function α ↦ rα, the authors establish that for any Brjuno α₀ with rα₀ > 0, there exist nearby parameters α arbitrarily close to α₀ with any prescribed smaller radius r < rα₀, and the associated uniformizing maps converge in smooth topology, yielding smooth boundary dynamics.

ABSTRACT

Recently, Xavier Buff and Arnaud Cheritat have provided an elegant proof of the existence of quadratic Siegel disks with smooth boundary. In this short note, we show how results of Yoccoz and Risler can be used to conclude the same result. Our proof is a small modification of the argument given by Buff and Cheritat.

Motivation & Objective

• To provide a simplified, abstract proof of the existence of quadratic Siegel disks with C∞ smooth boundaries, building on Buff and Cheritat's result.
• To demonstrate that the core dynamical mechanism relies not on parabolic explosion but on semicontinuity properties of the conformal radius function.
• To generalize the method beyond the quadratic family to other rational or entire maps without non-Brjuno Siegel disks.
• To extend the approach to Herman rings, showing analogous results for analytic linearization of circle diffeomorphisms.

Proposed method

• Utilizes upper semicontinuity of the conformal radius function rα, derived from Hurwitz's theorem and normal families.
• Applies weak lower semicontinuity of rα at both Brjuno and non-Brjuno irrationals, based on Yoccoz’s theorem (rα = 0 for non-Brjuno) and Risler’s result (continuity on dense sets of Brjuno numbers).
• Employs the Intermediate Value Theorem for functions that are both upper and weakly lower semicontinuous, to deduce density of parameters with any given radius r < rα₀.
• Constructs a sequence of parameters βi converging to α₀ such that rβi decreases monotonically to r, ensuring convergence of uniformizing maps in E_r topology.
• Uses the topology of uniform convergence on compact subsets and complete metric spaces E_r of smooth or quasianalytic functions to control regularity of the linearizing maps.
• Adapts the argument to Herman rings by replacing Yoccoz’s theorem with Geyer’s results on the complex Arnold family, preserving the semicontinuity structure.

Experimental results

Research questions

• RQ1 Can the existence of smooth Siegel disks in quadratic maps be established without relying on parabolic explosion techniques?
• RQ2 What minimal dynamical assumptions are required to ensure that the conformal radius function α ↦ rα satisfies the Intermediate Value Theorem?
• RQ3 To what extent can the method of semicontinuity and renormalization be generalized beyond quadratic polynomials?
• RQ4 Do similar results hold for Herman rings in rational maps, particularly in the complex Arnold family?
• RQ5 Is the weak lower semicontinuity of rα at non-Brjuno numbers sufficient to eliminate the need for special treatment of non-Brjuno parameters?

Key findings

• For any Brjuno α₀ with rα₀ > 0, and for any r < rα₀ and δ > 0, there exists α arbitrarily close to α₀ such that rα = r and the linearizing map Lα restricted to Dr is δ-close to Lα₀ in the E_r topology.
• The set of parameters α satisfying the conditions of Theorem 2.1 is a Cantor set, indicating uncountably many such smooth Siegel disks.
• The method applies to families of rational maps such as z ↦ e^{2πiα}z(1+z/d)^d for d ≥ 2, and z ↦ e^{2πiα}ze^z, provided they lack non-Brjuno Siegel disks.
• The same proof structure extends to Herman rings, yielding parameters λ near λ₀ with rλ = r < rλ₀ and Tλ|Ar close to Tλ₀|Ar in E_r.
• For Herman rings, the method also produces parameters λ with rλ = 0 but whose boundary conjugacy T is C⁰-close to the original Tλ₀|S¹.
• The proof shows that smoothness of the Siegel disk boundary is a consequence of topological and semicontinuity properties of the parameter space, not just dynamical control via parabolic explosion.