A transcendental Julia set of dimension 1

• 成果类型: Article
• 署名作者: Bishop, Christopher J.
• 署名单位: State University of New York (SUNY) System; Stony Brook University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0770-0
• 发表日期: 2018
• 页码: 407-460
• 关键词: hausdorff dimension; BOUNDARY; models; hairs

摘要

We construct a non-polynomial entire function whose Julia set has finite 1-dimensional spherical measure, and hence Hausdorff dimension 1. In 1975, Baker proved the dimension of such a Julia set must be at least 1, but whether this minimum could be attained has remained open until now. Our example also has packing dimension 1, and is the first transcendental Julia set known to have packing dimension strictly less than 2. It is also the first example with a multiply connected wandering domain where the dynamics can be completely described.