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.