Baker domains and singularities for certain meromorphic functions (Q5936405)

Let \(f\) be a meromorphic function. Let \(U\) be a periodic component of period \(p\) of the Fatou set of \(f\). Then \(U\) is called a Baker domain if \(f^{np}|_{U}\rightarrow z_{0}\) as \(n\rightarrow \infty\) for some \(z_{0}\in \partial U\) such that \(f^{p}(z_{0})\) is not defined. It is known that periodic cycles of Baker domains need not contain a singularity of \(f^{-1}\). Here it is shown that if \(f\) has only finitely many poles, then every cycle of multiply-connected Baker domain does contain a singularity of \(f^{-1}\).