On Siegel disks of a class of entire maps

Abstract

Let f : C → 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 quasicircle containing at least one critical point of f. This unifies and generalizes several previously known results.