Self-Similarity of Siegel Disks and Hausdorff Dimension of Julia Sets

Abstract

Let f(z) = e2 i z +z2, where θ is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: • the Hausdorff dimension of the Julia set J(f) is strictly less than two; and • if θ is a quadratic irrational (such as the golden mean), then the Siegel disk for f is self-similar about the critical point. In the latter case, we also show the rescaled first-return maps converge exponentially fast to a system of commuting branched coverings of the complex plane.