Xavier Buff and Arnaud Chéritat

2 Xavier Buff And

We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure in three cases: the presence of a Cremer fixed point, the presence of a Siegel disk, the presence of infinitely many (satellite) renormalizations.

The Solar Julia Sets of Basic Quadratic Cremer Polynomials

In general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impr...

S aling ratios and triangles in Siegel disks . byXavier Buff

Ghys-like Models for Lavaurs and Simple Entire Maps

We provide a new geometric construction of pre-models (à la Ghys) for Lavaurs maps, from which we deduce that their Siegel disk is a Jordan curve running through a critical point, which was not known before. The construction turns out to work also for a class of entire maps, very specific, nonetheless including cases where no pre-models were known.

The Brjuno Function and the Size of Siegel Disks

If α is an irrational number, we define Yoccoz's Brjuno function Φ by Φ(α) = n≥0 α 0 α 1 · · · α n−1 log 1 αn , where α 0 is the fractional part of α and α n+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 → e 2iπα z + z 2 has an indifferent fixed point at the origin. If Pα is linearizable, we let r(α) be the conf...