Boundary dynamics in unbounded Fatou components

TL;DR

This paper investigates the boundary structure of unbounded invariant Fatou components for transcendental entire functions, showing density of certain boundary points and measure-zero of singularities under specific conditions.

Contribution

It provides a detailed topological and measure-theoretic analysis of boundary dynamics in unbounded Fatou components, extending understanding of boundary behavior in complex dynamics.

Findings

Periodic and escaping boundary points are dense in the boundary.

All periodic boundary points are accessible from the Fatou component.

The set of singularities of the associated inner function has zero Lebesgue measure.

Abstract

We study the behaviour of a transcendental entire map $ f\colon \mathbb{C}\to\mathbb{C} $ on an unbounded invariant Fatou component $ U $, assuming that infinity is accessible from $ U $. It is well-known that $ U $ is simply connected. Hence, by means of a Riemann map $ \varphi\colon\mathbb{D}\to U $ and the associated inner function, the boundary of $ U $ is described topologically in terms of the disjoint union of clusters sets, each of them consisting of one or two connected components in $ \mathbb{C} $. Moreover, under more precise assumptions on the distribution of singular values, it is proven that periodic and escaping boundary points are dense in $ \partial U $, being all periodic boundary points accessible from $ U $. Finally, under the same conditions, the set of singularities of $ g $ is shown to have zero Lebesgue measure.