Bounded type Siegel disks of finite type maps with few singular values

Abstract

Let $U$ be an open subset of the Riemann sphere $\S$.We give sufficient conditions for which a finite type map $f:U\to~\S$ with at most three singular values has a Siegel disk compactly contained in $U$ and whose boundary is a quasicircle containing a unique critical point. The main tool is quasiconformal surgery ala Douady-Ghys-Herman-§w. We also give sufficient conditions for which, instead, $\Delta$ has not compact closure in $U$. The main tool is the Schwarzian derivative and area inequalities ala Graczyk-§w.