Siegel Disks and Periodic Rays of Entire Functions

Abstract

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk such that f restricts to a homeomorphism of the boundary. We show that the Siegel disk is bounded. Using a result of Herman, we deduce that if additionally the rotation number of the Siegel disk is Diophantine, then its boundary contains a critical point of f. Suppose furthermore that all singular values of f lie in the Julia set. We prove that, if f has a Siegel disk $U$ whose boundary contains no singular values, then the condition that f is a homeomorphism of the boundary of U is automatically satisfied. We also investigate landing properties of periodic dynamic rays by similar methods.