On the Size of Quadratic Siegel Disks: Part I

TL;DR

This paper proves that for quadratic polynomials with an irrational rotation number, the sum of the Bruno series and the logarithm of the Siegel disk's conformal radius is bounded, confirming a conjecture in complex dynamics.

Contribution

It provides a new proof that the sum of the Bruno series and the log of the Siegel disk radius is bounded, extending Yoccoz's results and confirming a key conjecture.

Findings

Established the boundedness of B(α)+log r(α) for all irrational α with finite Bruno series.

Provided a new proof technique for the relation between Bruno series and Siegel disk size.

Confirmed the conjecture linking arithmetic properties of α to the geometry of Siegel disks.

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 $P_\a:z\mapsto e^{2i\pi \a}z+z^2$ 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 $B(\a)<\infty$, we have $$B(\a)+\log r(\a) < C.$$ Together with former results of Yoccoz (see \cite{y}), this proves the conjectured boundedness of $B(\a)+\log r(\a)$.