Boundary dynamics in unbounded Fatou components

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.