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

Author(s)

Lomonaco, Luna ^[1] ; Petersen, Carsten Lunde ^[2]

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Dept Matemat Aplicada, Sao Paulo - Brazil

^[2] Roskilde Univ, INM, DK-4000 Roskilde - Denmark

• Document type: Journal article
• Source: INVENTIONES MATHEMATICAE; v. 210, n. 2, p. 615-644, NOV 2017.
• Web of Science Citations: 0

Abstract

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 . (AU)

• FAPESP's process: 13/20480-7 - Parabolic phenomena in complex dynamics
• Grantee: Luciana Luna Anna Lomonaco
• Support Opportunities: Scholarships in Brazil - Post-Doctoral