报告人： JUN HU 教授
Department of MathematicsGraduate Center and Brooklyn CollegeThe City Univ. of New York, USA
NYU-ECNU Institute of Mathematical SciencesNew York University - Shanghai, China
时间：5月3日 15：00-16：00
地点：数学楼二楼 报告厅
Abstract. Let Cb be the Riemann sphere. A rational map f : Cb → Cb is saidto be regularly ramified if for every point q ∈ Cb, all pre-images of q under fhave equal indices (meaning that f has same local degrees at all pre-imagesof q). Up to conjugacy by a M¨obius transformation, any regularly ramifiedrational map f can be written as a quotient map of a finite Kleinian grouppost-composed by a M¨obius transformation, and f can have only two or threecritical values.We explore and classify the Julia sets of such maps in some one-parameterfamilies fλ, where λ is a complex parameter. The maps in these families havea common super attracting fixed point of order = 2 or > 2. We show theyhave classifications similar to the classifications of the Julia sets of maps inthe families fcn(z) = zn + czn , where n is a positive integer = 2 or > 2 and cis a complex number. A new type of Julia set is also presented, which has notappeared in the literature and which are called exploded McMullen necklaces.We first prove that none of the maps in these families can have Herman ringsin their Fatou sets. Then we prove: if the super attracting fixed point of fλhas order greater than 2, then the Julia set J(fλ) is either connected, a Cantorset, or a McMullen necklace (either exploded or not); if the super attractingfixed point has order equal to 2, then J(fλ) is either connected or a Cantorset.Furthermore, we prove that the result on the nonexistence of Herman ringactually holds for every regularly ramified rational map.These are joint works with Francisco G. Jim´enez-L´opez, Oleg Muzician andYingqing Xiao.