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

 Title: Self-Similarity of Siegel Disks and Hausdorff Dimension of Julia Sets Author: McMullen, Curtis T. Citation: McMullen, Curtis T. 1998. Self-similarity of Siegel disks and the Hausdorff dimension of Julia sets. Acta Mathematica 180(2): 247–292. Revised 2004. Full Text & Related Files: McMullen_SelfSimilaritySiegelDisk.pdf (697.2Kb; PDF) 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. Published Version: doi:10.1007/BF02392901 Other Sources: http://www.math.harvard.edu/~ctm/papers/index.html Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3445997 Downloads of this work: