On the boundary of an immediate attracting basin of a hyperbolic entire function

TL;DR

This paper investigates the boundary structure of attracting basins for certain transcendental entire functions, showing under specific conditions that the intersection with the escaping set has Hausdorff dimension 1, generalizing previous results.

Contribution

It extends known results on the boundary of attracting basins from exponential functions to a broader class of transcendental entire functions with finite order.

Findings

The intersection of the boundary of the attracting basin with the escaping set has Hausdorff dimension 1.

The result applies to functions of the form $f(z)= extstylerac{1}{p(z)}rac{d}{dz}(p(z)e^{q(z)})+c$ with polynomials $p,q$.

Generalizes prior work on exponential functions to more complex transcendental entire functions.

Abstract

Let $f$ be a transcendental entire function of finite order which has an attracting periodic point $z_0$ of period at least $2$. Suppose that the set of singularities of the inverse of $f$ is finite and contained in the component $U$ of the Fatou set that contains $z_0$. Under an additional hypothesis we show that the intersection of $\partial U$ with the escaping set of $f$ has Hausdorff dimension $1$. The additional hypothesis is satisfied for example if $f$ has the form $f(z)=\int_0^z p(t)e^{q(t)}dt+c$ with polynomials $p$ and $q$ and a constant $c$. This generalizes a result of Bara\'nski, Karpi\'nska and Zdunik dealing with the case $f(z)=\lambda e^z$.