Accesses to infinity from Fatou components
  We study the boundary behaviour of a meromorphic map $f: \mathbb C \to \widehat{\mathbb C}$ on its invariant simply connected Fatou component $U$. To this aim, we develop the theory of accesses to boundary points of $U$ and their relation to the dynamics of $f$. In particular, we establish a correspondence between invariant accesses from $U$ to infinity or weakly repelling points of $f$ and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps. 
Krzysztof Bara\'nski, N\'uria Fagella, Xavier Jarque, Bogus{\l}awa   Karpi\'nska
math.DS