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Cylinder renormalization of Siegel disks. (English) Zbl 1145.37028

Summary: We study one of the central open questions in one-dimensional renormalization theory – the conjectural universality of golden-mean 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 a convincing evidence to support the hyperbolicity conjecture in this setting.

MSC:

37F25 Renormalization of holomorphic dynamical systems
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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