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

Autor(es)

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

Afiliação do(s) autor(es)

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

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

• Tipo de documento: Artigo Científico
• Fonte: INVENTIONES MATHEMATICAE; v. 210, n. 2, p. 615-644, NOV 2017.
• Citações Web of Science: 0

Resumo

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)

• Processo FAPESP: 13/20480-7 - Fenômenos parabólicos em dinâmica complexa
• Beneficiário: Luciana Luna Anna Lomonaco
• Modalidade de apoio: Bolsas no Brasil - Pós-Doutorado