On David type Siegel Disks of the Sine family

TL;DR

This paper extends Siegel surgery techniques to a broader class of functions without free critical points, demonstrating that for typical rotation numbers, the boundary of the Siegel disk is a Jordan curve passing through two critical points.

Contribution

It generalizes Petersen-Zakeri's Siegel surgery method to all premodels without free critical points, solving key questions about Siegel disk boundaries.

Findings

The boundary of the Siegel disk is a Jordan curve for typical rotation numbers.

The boundary passes through exactly two critical points.

The method applies to a wider class of functions than previously known.

Abstract

In 2008 Petersen posed a list of questions on the application of trans-quasiconformal Siegel surgery developed by Zakeri and himself. In this paper we extend Petersen-Zakeri's idea so that the surgery can be applied to all the premodels which have no "free critical points". We explain how the idea is used in solving three of the questions posed by Petersen. To present the details of the idea, we focus on the solution of one of them: we prove that for typical rotation numbers $0< \theta < 1$, the boundary of the Siegel disk of $f_{\theta}(z) = e^{2 \pi i \theta} \sin (z)$ is a Jordan curve which passes through exactly two critical points $\pi/2$ and $-\pi/2$.