Simply Connected Wandering Domains of Small Order Entire Functions

TL;DR

The paper proves that any bounded simply connected domain with analytic boundary can be realized as a wandering domain of an entire function of prescribed order α in (0,1), with univalent iterates on the domain. The construction blends Boc Thaler's approximation framework with Hörmander's ∂-theory, together with quantitative conformal straightening and carefully designed subharmonic weights to simultaneously control domain geometry and growth. It yields the first examples of such wandering domains for all α in (0,1), bridging growth-rate and geometric control previously treated separately. The method provides a structural blueprint for embedding prescribed geometric dynamics into entire maps while maintaining precise growth bounds.

Abstract

We show that any bounded, simply connected domain with analytic boundary can be realised as a wandering domain of an entire function of any prescribed order in $(0, 1)$. Extending results of Boc Thaler, our construction simultaneously prescribes the domain and the exact order of the map. In particular, we produce the first examples of entire functions with bounded simply connected wandering domains of each order in $(0,1/2).$

Figures (3)

• Figure 1: Schematic of the construction in Lemma \ref{['lem:small_disks']} (not to scale). The transition region where $\bar{\partial}\chi\neq 0$ is confined to the two thin annuli $\mathcal{A}_1$ and $\mathcal{A}_2$. The darkest inner disk bounded by $\mathcal{A}_1$ is mapped by the model map to a neighbourhood of $\tau$, while the small disks are mapped to a neighbourhood of $-A$. The subharmonic weight, $u$, is obtained by gluing across $\mathcal{A}_4$.
• Figure 2: First step: straightening using the Riemann map $\varphi_U^{-1}$, the domains are straightened so that $\partial U$ becomes circular and collars become annuli. The dark grey region indicates $\operatorname{supp}(\bar{\partial}\chi)$, confined to the thin annuli, $\mathcal{A}_1,\mathcal{A}_2$, and a narrow collar, $U_0\setminus V_1$. The subharmonic weight, $u_1$, is very negative on the regions where we require accurate approximation (namely on $B(\pm\tau_1,4)$ and on $\overline{U_1}$), while outside these sets it behaves essentially like $|z|^{\alpha}$. Thus the gluing occurs where $\bar{\partial}\chi_1\neq 0$ (the dark areas in the figure), which is where $u_1$ is large, away from the approximation zones where $u_1$ is small and controls the contribution of the $\bar{\partial}$–error.
• Figure 3: Schematic of the inductive step of the construction (not to scale). Inside the disk $B\!\left(0,\tfrac{10}{36}\tau_k\right)$, the map $f_k$ agrees closely with $f_{k-1}$, which sends the first $k-1$ forward images of $U_{k+1}$ into a set close to a round disk centred at $\tau_k$. The model map $h_k$, that $f_k$ approximates, sends $\overline{U}_{k+1}$ to a disk centred at $\tau_{k+1}$, while disks centred at the points $f^{\,k}_{k-1}(a_j^k)$ are mapped into a neighbourhood of $(-\tau_1)$. In this step we approximate these two behaviours by implementing the two maps just described near $0$ and near $\tau_k$, respectively using the puncture lemma to obtain precise control near their centres (where the relevant sets lie), and then gluing them with a cut-off function, $\chi$, for which $\nabla\chi$ is supported in thin annuli around the corresponding disks.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1: Hörmander
• Proposition 2.2
• Proposition 2.3
• proof
• Theorem 2.4: Ransford
• Corollary 2.5
• Lemma 2.6: The Puncture Lemma Ganny2025
• Lemma 2.7: Warschawski, War50