Prime ends and accesses to infinity from Fatou components

Abstract:

Let $f$ be a transcendental entire function and $U$ an unbounded simply connected Fatou component that is invariant under $f$. We look at the theory of prime ends and, in particular, we focus on different types of prime ends in $U$. These prime ends are related through a Riemann map to the points lying on the unit circle. Our aim is to investigate the connection between the behaviour of $f$ in $U$, whenever infinity is or is not accessible from $U$, and the boundary points of the unit disc.