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 $f(z)=\int_0^z p(t)e^{q(t)}dt+c$ with polynomials $p$ and $q$ and a constant $c$. This generalizes a result of Bara\'nski, Karpi\'nska and Zdunik dealing with the case $f(z)=\lambda e^z$.