[Paper Review] On the Size of Quadratic Siegel Disks: Part I

Xavier Buff, Arnaud Chéritat|ArXiv.org|May 5, 2003

Mathematical Dynamics and Fractals7 references3 citations

TL;DR

This paper provides a new proof of the boundedness of the sum $ B(\alpha) + \log r(\alpha) $, where $ B(\alpha) $ is the Bruno sum from continued fraction approximants and $ r(\alpha) $ is the conformal radius of the Siegel disk for the quadratic polynomial $ P_\alpha(z) = e^{2\pi i\alpha}z + z^2 $. The authors establish a uniform upper bound of 16 for this sum when $ B(\alpha) < \infty $, confirming a conjecture by Yoccoz and refining earlier estimates via holomorphic motion and hyperbolic geometry techniques.

ABSTRACT

If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2iπ\a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all irrational number $\a$ with $B(\a)

Motivation & Objective

• To provide a new, quantitative proof of the boundedness of $ B(\alpha) + \log r(\alpha) $, a key conjecture in Siegel disk theory.
• To refine the understanding of the conformal radius $ r(\alpha) $ of the Siegel disk for quadratic polynomials with rotation number $ \alpha $.
• To establish a uniform upper bound on $ B(\alpha) + \log r(\alpha) $ for all Bruno numbers $ \alpha $, independent of the continued fraction growth rate.
• To strengthen Yoccoz's earlier results by quantifying the decay of the Siegel disk radius in relation to the Bruno sum.

Proposed method

• The proof uses holomorphic motion techniques to track the evolution of periodic cycles near parabolic points as $ \alpha $ varies near rational approximants $ p_n/q_n $.
• It constructs a sequence of domains $ V_N $, defined as the complement of external rays and periodic points of period $ \leq q_N $, to contain the Siegel disk.
• The conformal radius of $ V_N $ is estimated using hyperbolic geometry, particularly the Schwarz lemma and distortion estimates in punctured disks.
• A key step involves lifting holomorphic motions to a universal cover and applying a universal bound on the hyperbolic distance in $ \mathbb{D} \setminus \{0\} $.
• The authors use the recurrence relation $ q_n = a_n q_{n-1} + q_{n-2} $ to compare $ q_n $ with Fibonacci numbers $ F_n $, enabling summation estimates.
• They apply a comparison lemma showing $ \log(\lambda q_n^2)/q_n \leq \log(\lambda F_n^2)/F_n $ for $ \lambda > 81/64 $, which helps control the growth of the Bruno sum.

Experimental results

Research questions

• RQ1 Is the sum $ B(\alpha) + \log r(\alpha) $ uniformly bounded for all irrational $ \alpha $ with $ B(\alpha) < \infty $?
• RQ2 Can a new, quantitative proof be given for the boundedness of $ B(\alpha) + \log r(\alpha) $, independent of Yoccoz's original approach?
• RQ3 What is the optimal universal upper bound for $ B(\alpha) + \log r(\alpha) $, and can it be explicitly computed?
• RQ4 How does the conformal radius $ r(\alpha) $ decay as the Bruno sum $ B(\alpha) $ increases, and is this decay controlled uniformly across all $ \alpha $?
• RQ5 Can the dynamics of periodic cycles near parabolic points be used to estimate the size of the Siegel disk?

Key findings

• The sum $ B(\alpha) + \log r(\alpha) $ is uniformly bounded above by 16 for all irrational $ \alpha $ with $ B(\alpha) < \infty $, confirming the conjectured boundedness.
• The conformal radius $ r(\alpha) $ of the Siegel disk satisfies $ \log r(\alpha) < 16 - B(\alpha) $ whenever $ B(\alpha) < \infty $, providing a sharp quantitative estimate.
• The proof establishes that $ \log \text{rad}(V_N) + \sum_{n=0}^N \frac{\log q_{n+1}}{q_n} < 16 $, where $ V_N $ is the domain excluding periodic points of period $ \leq q_N $ and the external ray of argument 0.
• The Siegel disk, if it exists, is contained in the intersection of all $ V_N $, so the bound on $ \text{rad}(V_N) $ implies the bound on $ r(\alpha) $.
• The authors show that the constant 16 is not optimal, but the proof does not aim to minimize it, focusing instead on establishing a finite universal bound.
• The method improves on earlier estimates by incorporating a refined distortion control in the holomorphic motion framework, particularly via the use of $ \lambda $-maps and hyperbolic geometry in punctured disks.