On the connectivity of the Julia sets of meromorphic functions

• 成果类型: Article
• 署名作者: Baranski, Krzysztof; Fagella, Nuria; Jarque, Xavier; Karpinska, Boguslawa
• 署名单位: University of Warsaw; University of Barcelona; Warsaw University of Technology
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0504-5
• 发表日期: 2014
• 页码: 591-636
• 关键词: newtons method; analytic-functions; baker domains; Iteration; DYNAMICS; MAPS

摘要

We prove that every transcendental meromorphic map f with disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions, whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton's method for entire maps are simply connected, which solves a well-known open question.