Self-similarity of Siegel disks and Hausdorff dimension of Julia sets

Let f(z) = e 2ßi` z + z 2 , where ` is an irrational number of bounded type. According to Siegel, f is linearizable on a disk containing the origin. In this paper we show: ffl the Hausdorff dimension of the Julia set J(f) is strictly less than two; and ffl if ` 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. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Rotations of the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Visiting the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Geometry of the Julia set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2..