Rotation Domains and Stable Baker Omitted Value

Abstract

A Baker omitted value, in short bov of a transcendental meromorphic function f is an omitted value such that there is a disk D centered at the bov for which each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming all the iterates \(f^n\) to be analytic in a neighborhood of its bov, this article proves a number of results on the rotation domains of the function. The number of p-periodic Herman rings is shown to be finite for each \(p \ge 1\). It is also proved that every Julia component intersects the boundaries of at most finitely many Herman rings. Further, if the bov is the only limit point of the critical values then it is shown that f has infinitely many repelling fixed points. If a repelling periodic point of period p is on the boundary of a p-periodic rotation domain then the periodic point is shown to be on the boundary of infinitely many Fatou components. As a corollary we have shown that if D is a p-periodic rotation domain of f and \(f^p\) is univalent in a neighbourhood of the boundary \( \partial D \) of D then f has no repelling p-periodic point on \( \partial D \). Under additional assumptions on the critical points, a sufficient condition is found for a Julia component to be a singleton. As a consequence, it is proved that if the boundary of a wandering domain W accumulates at some point of the plane under the iteration of f then each limit of \(f^n\) on W is either a parabolic periodic point or in the \(\omega \)-limit set of recurrent critical points. Using the same ideas, the boundaries of rotation domains are shown to be in the \(\omega \)-limit set of recurrent critical points.