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

We study the dimension of the boundaries of periodic Fatou components of transcendental entire maps. We prove that if $U$ is an immediate component of the basin of an attracting periodic point $\zeta$ of period $p\ge 1$ of a transcendental entire function $f\colon \mathbb C \to \mathbb C$ from the Eremenko--Lyubich class $\mathcal B$, such that $\text{deg} f^p|_U = \infty$ and $\overline{\text{Sing}(f^p|_U)}$ is a compact subset of $U$, then the hyperbolic (and, consequently, Hausdorff) dimension of the boundary of $U$ is larger than $1$. The same holds if $U$ is an immediate component of the basin of a parabolic $p$-periodic point $\zeta$, under an additional assumption $\zeta \notin \overline{\text{Sing}(f^p)}$. We also show that if $U$ is a bounded immediate component of an attracting basin of a transcendental entire function $f$, then the hyperbolic dimension of the boundary of $U$ is larger than $1$. In particular, this implies that the boundary of a component of an attracting basin of a transcendental entire function is never a smooth or rectifiable curve.