Escaping points in the boundaries of Baker domains

TL;DR

This paper investigates the boundary behavior of points in Baker domains of meromorphic functions, showing that the type of the domain determines whether boundary points escape to infinity or have dense trajectories.

Contribution

It classifies boundary dynamics in Baker domains based on the function's type, extending results to infinite degree cases including classical examples.

Findings

Almost all boundary points escape to infinity in hyperbolic or simply parabolic types.

Boundary points have dense trajectories in doubly parabolic type.

Results extend to infinite degree cases, including classical Fatou examples.

Abstract

We study the dynamical behaviour of points in the boundaries of simply connected invariant Baker domains $U$ of meromorphic maps $f$ with a finite degree on $U$. We prove that if $f|_U$ is of hyperbolic or simply parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure escapes to infinity under iteration. On the contrary, if $f|_U$ is of doubly parabolic type, then almost every point in the boundary of $U$ with respect to harmonic measure has dense forward trajectory in the boundary of $U$, in particular the set of escaping points in the boundary of $U$ has harmonic measure zero. We also present some extensions of the results to the case when $f$ has infinite degree on $U$, including classical Fatou example.