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

TL;DR

The paper proves that for transcendental entire maps in class $\mathcal B$, boundaries of attracting or parabolic basin components typically have hyperbolic dimension greater than 1 under natural infinite-degree conditions, and even for bounded immediate components. The authors construct expanding conformal repellers inside basin boundaries by dual logarithmic lifts and a carefully designed iterated function system, applying Bowen’s formula to obtain a dimension lower bound. These results imply that such boundaries are never smooth or rectifiable, addressing a question from Hayman’s list and providing a partial answer in non-polynomial dynamics. The methods unify topological, geometric, and thermodynamic tools to understand the fractal geometry of Fatou component boundaries in the transcendental setting.

Abstract

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a transcendental entire map from the Eremenko-Lyubich class $\mathcal{B}$, and let $ζ$ be an attracting periodic point of period $p$. We prove that the boundaries of components of the attracting basin of (the orbit of) $ζ$ 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 $ζ$, under the additional assumption $ζ\notin \overline{\text{Sing}(f^p)}$. 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.

Figures (10)

• Figure 1: The three types of periodic components of attracting basins of entire transcendental maps. Left: an unbounded periodic component $U$ of period $3$ with $\deg f^3|_U = \infty$ for $f(z) = (2+\pi i) e^z$ (see bhattacharjee-devaney00). Center: an unbounded invariant component $U$ with $\deg f|_U = 2$ for $f(z)=e^{-z}+z-1$ (see baker-dominguezweinreich). Right: a bounded invariant component $U$ with $\deg f|_U = 2$ for $f(z)= (-21+3i) z^2e^z$ (see garijo-jarque-morenorocha10).
• Figure 2: Topological structure of the component $U$.
• Figure 3: Logarithmic lift on $U \setminus E$.
• Figure 4: The sets $T_{s_0}^{(s_1)}$ and curves $\eta_{s_0, s_1}$.
• Figure 5: The rectangle $K$.

Theorems & Definitions (40)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Proposition 2.1: bergweiler-fagella-rempegillen15
• Lemma 4.1
• proof
• Lemma 5.1
• Lemma 6.1: Existence of the curve $\alpha$ and points $v_s$
• proof
• Lemma 6.2