Dynamics and Renormalization Seminar
Friday May 14th, 2021
Time: 10:00 AM - 11:00 AM
Title: Local connectivity of the Julia sets of holomorphic maps with bounded type Siegel disks
Speaker: Fei Yang , Nanjing University
Abstract:
Let f be a holomorphic map containing an irrationally indifferent fixed point z0. If f is locally linearizable at z0, then the maximal linearizable domain containing z0 is called the Siegel disk of f centered at z0. The topology of the boundaries of Siegel disks has been studied extensively in past 3 decades. This was motivated by the prediction of Douady and Sullivan that the Siegel disk of every non-linear rational map is a Jordan domain.

For the topology of whole Julia sets of holomorphic maps with Siegel disks, the results appear less. Petersen proved that the quadratic Julia sets with bounded type Siegel disks are locally connected. Later Yampolsky proved the same result by an alternative method based on the existence of complex bound of unicritical circle maps. A big progress was made by Petersen and Zakeri in 2004. They proved that for almost all rotation number, the quadratic Julia sets with Siegel disks are locally connected. Recently J. Yang proved a striking result that the Julia set of any polynomial (assumed to be connected) is locally connected at the boundary points of their bounded type Siegel disks.

In this talk we prove that a long iteration of a class of quasi-Blaschke models has certain expanding property near the unit circle. This leads us to prove the local connectivity of the Julia sets of a number of rational maps and transcendental entire functions with bounded type Siegel disks. This is a joint work with S. Wang, G. Zhang and Y. Zhang.