On Siegel disks of a class of entire maps

Abstract

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire map of the form $f(z)=P(z)\exp (Q(z))$, where $P$ and $Q$ are polynomials of arbitrary degrees. (We allow the case $Q=0$.) Building upon a method pioneered by M. Shishikura, we show that if $f$ has a Siegel disk of bounded-type rotation number centered at the origin, then the boundary of this Siegel disk is a quasi circle containing at least one critical point of $f$. This unifies and generalizes several previously known results