Boundary dynamics in unbounded Fatou components

Abstract

We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected. Hence, by means of a Riemann map $ \varphi\colon\mathbb{D}\to U $ and the associated inner function, the boundary of $ U $ is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in $ \mathbb{C} $. Moreover, under more precise assumptions on the distribution of singular values, it is proven that periodic and escaping boundary points are dense in $ \partial U $, being all periodic boundary points accessible from $ U $. Finally, under the same conditions, the set of singularities of $ g $ is shown to have zero Lebesgue measure.