Siegel Disks and Periodic Rays of Entire Functions

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk U such that f |∂U is a homeomorphism. We extend a theorem of Herman by showing that, if the rotation number of U is diophantine, then ∂U contains a critical point of f . More generally, we show that, if U is a (not necessarily diophantine) Siegel disk as above, then U is bounded. 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 : ∂U → ∂U is a homeomorphism is automatically satisfied. We also investigate landing properties of periodic dynamic rays by similar methods.