A model for boundary dynamics of Baker domains

Abstract

We consider the transcendental entire function $f(z)=z+e^{-z}$, which has a doubly parabolic Baker domain $U$ of degree two, i.e. an invariant stable component for which all iterates converge locally uniformly to infinity, and for which the hyperbolic distance between successive iterates converges to zero. It is known from general results that the dynamics on the boundary is ergodic and recurrent and that the set of points in $\partial U$ whose orbit escapes to infinity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, and it is organized in curves encoded by some symbolic dynamics, whose closure is precisely $\partial U$. We also prove that nevertheless, all escaping points in $\partial U$ are non-accessible from $U$, as opposed to points in $\partial U$ having a bounded orbit, which are all accessible. Moreover, repelling periodic points are shown to be dense in $\partial U$, answering a question posted in (Barański et al. in J Anal Math 137:679-706, 2019). None of these features are known to occur for a general doubly parabolic Baker domain.