Let be a function which is analytic in a neighborhood of with and (where R\Q). The famous

In the case where is an entire function and is linearizable, is the center of a *Siegel disk*,
; that
is, a maximal domain
on which is linearizable. It is an important question
to determine
when the boundary of contains a singular value of .

Herman proved the following result: if the rotation number is of diophantine type, the Siegel disk is bounded and is a homeomorphism, then contains a critical point. While this result is rather satisfactory when is a polynomial, it is far less complete for transcendental entire functions. We will show that, if is a transcendental entire function whose set of singular values is bounded, then the assumption that is bounded can be removed. This is a considerable improvement of Herman's result in this case, and in particular answers a question of Herman, Baker and Rippon.