On the scaling ratios for Siegel disks

TL;DR

This paper investigates the self-similarity and scaling ratios of Siegel disks for quadratic polynomials with specific irrational rotation numbers, providing bounds and estimates for the scaling ratio based on the continued fraction expansion.

Contribution

It offers explicit bounds on the scaling ratio of Siegel disks and estimates the quasisymmetric constant of the conjugacy, advancing understanding of their universal geometric properties.

Findings

Bounds on the scaling ratio  in terms of continued fraction period s

Explicit estimate of the quasisymmetric constant of the conjugacy

Universal geometric bounds for a class of holomorphic maps

Abstract

The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self-similar with a certain scaling ratio. The restriction of the dynamics of the quadratic polynomial to the boundary of the Siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number. 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. In this paper we provide an estimate on the quasisymmetric constant of the conjugacy, and use it to prove bounds on the scaling ratio $\lambda$ of the form $$\alpha^\gamma \le |\lambda| \le C \delta^s,$$ where $s$ is the period of the continued fraction, and $\alpha \in (0,1)$ depends on the rotation number in an explicit way, while $C>1$, $\delta \in (0,1)$ and $\gamma \in (0,1)$ depend only on the maximum of the integers in the continued fraction expansion of the rotation number.