Accesses to infinity from Fatou components

TL;DR

This paper investigates the boundary behavior of meromorphic functions on invariant Fatou components, establishing a connection between accesses to boundary points and the function's dynamics, with applications to Newton maps.

Contribution

It develops a new theory of accesses to boundary points of Fatou components and links these accesses to boundary fixed points of associated inner functions.

Findings

Established a correspondence between invariant accesses and boundary fixed points.

Described accesses to infinity in invariant Fatou components of Newton maps.

Connected boundary behavior with the dynamics of meromorphic functions.

Abstract

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.