Boundaries of Baker domains of entire functions. A finer approach

TL;DR

This paper analyzes the boundary dynamics of doubly parabolic Baker domains in transcendental entire functions, providing a measure-theoretic description and improving existing models, especially for the function z+e^{-z}.

Contribution

It introduces new results on the boundary dynamics of inner functions related to Baker domains, refining previous models and extending understanding of their topological and measure-theoretic properties.

Findings

Describes boundary dynamics of doubly parabolic Baker domains measure-theoretically.

Improves existing models for the Baker domain of z+e^{-z}.

Establishes new results on the radial extension of doubly parabolic inner functions.

Abstract

We consider transcendental entire functions having doubly parabolic Baker domains, such that the Denjoy-Wolff point of the associated inner function is not a singularity. We describe in a very precise way the dynamics on the boundary from a measure-theoretical point of view. Applications of such results lead to a better understanding of the topology and the dynamics on the boundaries. In particular, we improve some of the results in [N. Fagella and A. Jov\'e, A model for boundary dynamics of Baker domains], for the Baker domain of $z+e^{-z}$. In fact, our conclusions are obtained by applying new results established here on the dynamics of the radial extension of one component doubly parabolic inner functions, which strengthen those of [O. Ivrii and M. Urba\'nski, Inner functions, composition operators, symbolic dynamics and thermodynamic formalism].