seminar

Let $f: \mathbb{C} \to \widehat{\mathbb{C}}$ be a meromorphic function (analytic everywhere except at poles) with a single essential singularity. The Fatou set of $f$, $\mathcal{F}(f)$is the subset of the plane where $\{f^n\}_{n >0}$ forms a normal family. A component $H$ of $\mathcal{F}(f)$ is called a Herman ring if there exists an analytic homeomorphism $\phi: \{z: 1<|z|< R\} \to H$ such that $\phi^{-1}(f^p (\phi(z)))$ is an irrational rotation about the origin for some natural number $p$. This $p$ is called the period of the Herman ring $H$. A component of the Fatou set is called completely invariant if $f(U) \subseteq U$ and $f^{-1}(U) \subseteq U$. If $f$ has an omitted value then it is shown that $f$ has no Herman ring of period one or two and the number of completely invariant components of $\mathcal{F}(f)$ is at most two (except for a restricted case).