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

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 $f\left(z\right)={\int }_0^{z}p\left(t\right)e^{q\left(t\right)}dt+c$ with polynomials $p$ and $q$ and a constant $c$. This generalizes a result of Barański, Karpińska, and Zdunik dealing with the case $f\left(z\right)=\lambda e^z$.