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

TL;DR

This paper proves local connectivity of Julia sets for certain transcendental entire functions with Siegel disks, including sine functions with bounded type rotation numbers, using properties of quasi-Blaschke products.

Contribution

It establishes local connectivity of Julia sets for a class of transcendental entire functions with Siegel disks, extending previous results to functions with asymptotic values.

Findings

Julia sets are locally connected for functions with bounded type Siegel disks

Sine functions with bounded type rotation numbers have locally connected Julia sets

Existence of transcendental entire functions with Siegel disks and locally connected Julia sets with asymptotic values

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 $\theta$ is of bounded type, then the Julia set of the sine function $S_\theta(z)=e^{2\pi i\theta}\sin(z)$ is locally connected. Moreover, we prove the existence of transcendental entire functions having Siegel disks and locally connected Julia sets with asymptotic values.