Cylinder renormalization for Siegel disks and a constructive Measurable Riemann Mapping Theorem

Abstract

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this explanation, the hyperbolicity of renormalization, has not been proved yet. The present work considers a cylinder renormalization - a novel type of renormalization for holomorphic maps with a Siegel disk which is better suited for a hyperbolicity proof. A key element of a cylinder renormalization of a holomorphic map is a conformal isomorphism of a dynamical quotient of a subset of $\field{C}$ to a bi-infinite cylinder $\field{C} / \field{Z}$. A construction of this conformal isomorphism is an implicit procedure which can be performed using the Measurable Riemann Mapping Theorem. We present a constructive proof of the Measurable Riemann Mapping Theorem, and obtain rigorous bounds on a numerical approximation of the desired conformal isomorphism. Such control of the uniformizing conformal coordinate is of key importance for a rigorous computer-assisted study of cylinder renormalization.