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The Brjuno function continuously estimates the size of quadratic Siegel disks. (English) Zbl 1109.37040

Authors’ abstract: If \(\alpha\) is an irrational number, Yoccoz defined the Brjuno function \(\Phi\) by \[ \Phi(\alpha)=\sum_{n\geq 0}\alpha_0\alpha_1\cdots\alpha_{n-1}\log\frac{1}{\alpha_n}, \] where \(\alpha_0\) is the fractional part of \(\alpha\) and \(\alpha_{n+1}\) is the fractional part of \(1/\alpha_n\). The numbers \(\alpha\) such that \(\Phi(\alpha)<+\infty\) are called the Brjuno numbers. The quadratic polynomial \(P_\alpha:z\mapsto e^{2i\pi\alpha}z+z^2\) has an indifferent fixed-point at the origin. If \(P_\alpha\) is linearizable, we let \(r(\alpha)\) be the conformal radius of the Siegel disk and we set \(r(\alpha)=0\) otherwise. J.-C. Yoccoz [Small divisors in dimension one, Astérisque, 231, Paris: Société Mathématique de France (1995; Zbl 0836.30001)] proved that \(\Phi(\alpha)=+\infty\) if and only if \(r(\alpha)=0\) and that the restriction of \(\alpha\mapsto \Phi(\alpha)+\log r(\alpha)\) to the set of Brjuno numbers is bounded from below by a universal constant. In [the authors, Invent. Math. 156, 1–24 (2004; Zbl 1087.37041)], we proved that it is also bounded from above by a universal constant. In fact, S. Marmi, P. Moussa and J.-C. Yoccoz [Commun. Math. Phys. 186, No. 2, 265–293 (1997; Zbl 0947.30018)] conjecture that this function extends to \(\mathbb{R}\) as a Hölder function of exponent \(1/2\).
In this article, we prove that there is a continuous extension to \(\mathbb{R}\).
Reviewer: Pei-Chu Hu (Jinan)

MSC:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
30C10 Polynomials and rational functions of one complex variable
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30B70 Continued fractions; complex-analytic aspects
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