TAILIEUCHUNG - Đề tài " The Brjuno function continuously estimates the size of quadratic Siegel disks "

If α is an irrational number, Yoccoz defined the Brjuno function Φ by Φ(α) = n≥0 α0α1 · · · αn−1 log 1 αn , 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) | Annals of Mathematics The Brjuno function continuously estimates the size of quadratic Siegel disks By Xavier Buff and Arnaud Ch eritat Annals of Mathematics 164 2006 265 312 The Brjuno function continuously estimates the size of quadratic Siegel disks By Xavier Buff and Arnaud Chéritat Abstract If a is an irrational number Yoccoz defined the Brjuno function by a 52 O0 1 n-1 log -1 n 0 n where a0 is the fractional part of a and an 1 is the fractional part of 1 an. The numbers a such that a to are called the Brjuno numbers. The quadratic polynomial Pa z e2inaz z2 has an indifferent fixed point at the origin. If Pa is linearizable we let r a be the conformal radius of the Siegel disk and we set r a 0 otherwise. Yoccoz Y proved that a rc if and only if r a 0 and that the restriction of a a log r a 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 Holder function of exponent 1 2. In this article we prove that there is a continuous extension to R. Contents 1. Introduction 2. Statement of results . The value of Y at rational numbers . The value of Y at Cremer numbers . Strategy of the proof 3. Parabolic explosion . Outline . Definitions . A preliminary lemma Getting some room for holomorphic motions . The loss of conformal radius when one removes the exploding cycle . A short remark Denominators of convergents and Fibonacci numbers . The key inequality for the upper bound . Application to the proof of Theorem 2 Y at Cremer numbers 266 XAVIER BUFF AND ARNAUD CHERITAT 4. Proof of inequality 4 the upper bound . Irrational numbers . Rational numbers Outline . Rational numbers . Proof of Lemma 3 Removing external rays for a close to p q 5. Yoccoz s renormalization techniques . Outline . Renormalization principle . Proof of Propostion 3 The .

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