Local connectivity of Julia sets for rational maps with Siegel disks

Abstract

We prove that the Julia sets of a number of rational maps and transcendental entire functions with bounded type Siegel disks are locally connected. This is based on establishing an expanding property of a long iteration of a class of quasi-Blaschke models near the unit circle. Contents 1. Introduction 1 2. Quasi-Blaschke models and half hyperbolic neighborhoods 6 3. Admissible sequences and contraction regions 14 4. Proof of the Key Lemma 21 5. Proofs of Theorems A and B 25 References 31

A brief introduction to one-dimensional complex dynamics

“…通过类似的办法, 并基于退化 Beltrami 方程 的研究 (参见文献 [216]), Petersen 和 Zakeri [569] 将前者的结果推广到了几乎所有的无理数. 最近, 在 不依赖于拼图的情形下, Petersen 局部连通性的结果被推广到了一类含有界型 Siegel 盘的有理函数和 超越整函数 [748] .…”