On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

• 成果类型: Article
• 署名作者: Lomonaco, Luna; Petersen, Carsten Lunde
• 署名单位: Universidade de Sao Paulo; Roskilde University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0737-1
• 发表日期: 2017
• 页码: 615-644
• 关键词: levin-yoccoz inequality; mappings

摘要

Douady and Hubbard (Ann Sci Ec Norm Suppl 4 18(2):287-343, 1985) introduced the notion of polynomial-like maps. They used it to identify homeomorphic copies of the Mandelbrot set inside the Mandelbrot set . These copies can be primitive (with a root cusp) or satellite (without a root cusp). They conjectured that the primitive copies are quasiconformally homeomorphic to , and that the satellite ones are quasiconformally homeomorphic to outside any small neighbourhood of the root. These conjectures are now Theorems due to Lyubich (Ann Math 149:319-420, 1999). The satellite copies are clearly not q-c homeomorphic to . But are they mutually q-c homeomorphic? Or even q-c homeomorphic to half of the logistic Mandelbrot set? In this paper we prove that, in general, the induced Douady-Hubbard homeomorphism is not the restriction of a q-c homeomorphism: For any two satellite copies and the induced Douady-Hubbard homeomorphism is not q-c if the root multipliers and have .