A model for boundary dynamics of Baker domains

TL;DR

This paper analyzes the boundary dynamics of a specific transcendental entire function with a doubly parabolic Baker domain, revealing detailed structure of escaping points, symbolic organization, and accessibility properties, which are novel for such domains.

Contribution

The paper provides new insights into boundary dynamics of a particular Baker domain, including the organization of escaping points and the density of repelling periodic points.

Findings

Escaping set on the boundary is non-empty and organized by symbolic dynamics.

All escaping boundary points are non-accessible from the domain.

Repelling periodic points are dense in the boundary.

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 infity, 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 infity has zero harmonic measure. For this model we show that stronger results hold, namely that this escaping set is non-empty, 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 @U, answering a question posted Baranski, Fagella, Jarque and Karpinska. None of these features are known to occur for a general doubly parabolic Baker domain.