Cylinder Renormalization of Siegel Disks

Abstract

We study one of the central open questions in one-dimensional renormalization theory—the conjectural universality of goldenmean Siegel disks. We present an approach to the problem based on cylinder renormalization proposed by the second author. Numerical implementation of this approach relies on the constructive measurable Riemann mapping theorem proved by the first author. Our numerical study yields convincing evidence to support the hyperbolicity conjecture in this setting.

Keywords:

• Primary 37F25
• Secondary 30C62
• Siegel disc
• renormalization
• universality
• Beltrami equation
• Measurable Riemann Mapping Theorem