2 Xavier Buff And

S aling ratios and triangles in Siegel disks . byXavier Buff

Scaling ratios and triangles in Siegel disks

Let f (z) = e 2iπθ z + z 2 , where θ is a quadratic irrational. McMullen proved that the Siegel disk for f is self-similar about the critical point. We give a lower bound for the ratio of self-similarity, and we show that if θ = (√ 5 − 1)/2 is the golden mean, then there exists a triangle contained in the Siegel disk, and with one vertex at the critical point. This answers a 15 year old conject...

Xavier Buff and Arnaud Chéritat

We prove the existence of quadratic polynomials having a Julia set with positive Lebesgue measure. We find such examples with a Cremer fixed point, with a Siegel disk, or with infinitely many satellite renormalizations.

Geometry of the Feigenbaum Map

We show that the Feigenbaum-Cvitanović equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this basin which enables to recover a result of local connectivity by Jiang and Hu for the Feigenbaum Julia set.

Teichmüller spaces and holomorphic dynamics

One fundamental theorem in the theory of holomorphic dynamics is Thurston’s topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston’s theorem (the marked Thurston’s theorem). We also mention some applications and related results...