On the dimension of the boundaries of attracting basins of entire maps

Abstract

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $\zeta$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the attracting basin of (the orbit of) $\zeta$ have hyperbolic (and, consequently, Hausdorff) dimension larger than $1$, provided $f^p$ has an infinite degree on an immediate component $U$ of the basin, and the singular set of $f^p|_U$ is compactly contained in $U$. The same holds for the boundaries of components of the basin of a parabolic $p$-periodic point $\zeta$, under the additional assumption $\zeta \notin \overline{{\text{Sing}}(f^p)}$. We also prove that if an immediate component of an attracting basin of an arbitrary transcendental entire map is bounded, then the boundaries of components of the basin have hyperbolic dimension larger than $1$. This enables us to show that the boundary of a component of an attracting basin of a transcendental entire function is never a smooth or rectifiable curve. The results provide a partial answer to a question from Hayman's list of problems in function theory.