Local connectivity of Julia sets of some transcendental entire functions with Siegel disks

TL;DR

The paper addresses when Julia sets of certain transcendental entire functions with Siegel disks are locally connected, establishing a weak expansion framework via quasi-Blaschke models and hyperbolic orbifold metrics. This approach yields LC Julia sets for a broad class, including the sine family $S_\theta(z)=e^{2π i θ}\sin z$ with bounded-type $θ$, and it extends to functions with asymptotic values through polynomial-like renormalization. The combination of pullback control, half-hyperbolic neighborhood analysis, and Whyburn’s LC criterion provides a robust method to deduce LC of $J(f)$ in settings where uniform expansion is unavailable. These results deepen our understanding of the global dynamics of transcendental entire maps containing Siegel disks and offer constructive examples with LC Julia sets even when asymptotic values are present.

Abstract

Based on the weak expansion property of a long iteration of a family of quasi-Blaschke products near the unit circle established recently, we prove that the Julia sets of a number of transcendental entire functions with bounded type Siegel disks are locally connected. In particular, if $θ$ is of bounded type, then the Julia set of the sine function $S_θ(z)=e^{2πiθ}\sin(z)$ is locally connected. Moreover, we prove the existence of transcendental entire functions having Siegel disks and locally connected Julia sets with asymptotic values.

Figures (3)

• Figure 1: The dynamical plane of $S_\theta(z)=e^{2\pi\textup{i}\theta}\sin(z)$, where $\theta=(\sqrt{5}-1)/2$ is of bounded type (courtesy of A. Chéritat). The Siegel disk is colored yellow with invariant curves and the rest Fatou components are colored white. The Julia set (colored black and layered gray) of $S_\theta$ is locally connected.
• Figure 2: A half hyperbolic neighborhood $H_d(I)$ of the open arc $I$.
• Figure 3: Left: The parameter plane of $f_\lambda(z)=\lambda z e^z$, where the unit circle is marked by a yellow circle. Right: A zoom of the left picture near a Mandelbrot copy, which is inclosed by a yellow box. See also Fag95.

Theorems & Definitions (30)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Corollary 2.5