Boundaries of Baker domains of entire functions. A finer approach

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].