Siegel disks of entire functions

Let f : C -> C be an entire function with f(0) = 0 and |f'(0)|=1. We say that 0 is linearizable if there exists a neigborhood of 0 on which f is conjugate to the rotation z -> f'(0)*z. (The question under which conditions this is the case has become famous as the "center problem", with contributions by Siegel, Cremer, Bryuno, Yoccoz and others.)

If 0 is linearizable, then the maximal domain of linearization U is called a Siegel disk. It is an important question under which conditions U has a critical point of f on its boundary.

The main result on this question was proved in the '80s by Herman, who showed the following. Suppose that the rotation number of f at 0 is of diophantine type. Then, provided that the Siegel disk U is bounded and f is a homeomorphism when restricted to the boundary of U, the boundary of U contains a critical point. This result provides a very good answer in the case of polynomials, since in this case Siegel disks are always bounded. However, the answer is far less satisfactory when f is allowed to be transcendental.

We will prove the following: if the set of singular values of f is bounded and f is injective on the boundary of the Siegel disk U, then U is always bounded. In particular, Herman's result implies that under these conditions the boundary contains a critical point provided that the rotation number is diophantine. A special case of our theorem answers a question posed by Herman, Baker and Rippon in 1989.

Lasse Rempe

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