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

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.