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.

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