The Brjuno function continuously estimates the size of quadratic Siegel disks | Annals of Mathematics

Abstract If α is an irrational number, Yoccoz defined the Brjuno function Φ by Φ(α)=∑n≥0α0α1⋯αn−1log1αn, where α0 is the fractional part of α and αn+1 is the fractional part of 1/αn. The numbers α such that Φ(α)<+∞ are called the Brjuno numbers. The quadratic polynomial Pα:z↦e2iπαz+z2 has an indifferent fixed point at the origin. If Pα is linearizable, we let r(α) be the conformal radius of the Siegel disk and we set r(α)=0 otherwise. Yoccoz [Y] proved that Φ(α)=+∞ if and only if r(α)=0 and that the restriction of α↦Φ(α)+logr(α) to the set of Brjuno numbers is bounded from below by a universal constant. In [BC2], we proved that it is also bounded from above by a universal constant. In fact, Marmi, Moussa and Yoccoz [MMY] conjecture that this function extends to ℝ as a Hölder function of exponent 1/2. In this article, we prove that there is a continuous extension to ℝ.

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数学年刊(Annals of Mathematics)

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