Accessible fixed points on the boundary of stable domains
Let \(U\) be an invariant component of the Fatou set of a rational function \(R\) and \(z_0\in\partial U\) a weakly repelling fixed point of \(R\). We say that \(R\) is locally surjective at \(z_0\) with respect to \(U\) if, for any sufficiently small neighbourhood \(N\) of \(z_0\) we have \(R(N\cap U) =R(N) \cap R(U)\). This local surjectivity condition is certainly satisfied in the special case when \(U\) is completely invariant. The main result is that if \(R\) is locally surjective at the weakly repelling fixed point \(z_0 \in\partial U\) then \(z_0\) is an accessible boundary point. The result is due in this form to \textit{F. Prztycki} [Fundam. Math. 144, 259-278 (1994; Zbl 0812.58058)], but the present simpler proof refines a treatment of the case of simply-connected \(U\) given by \textit{L. Carleson} and \textit{T. W. Gamelin}, Complex dynamics (1993; Zbl 0782.30022).
scientific article; zbMATH DE number 1112243