Mating quadratic maps with the modular group II

• 成果类型: Article
• 署名作者: Bullett, Shaun; Lomonaco, Luna
• 署名单位: University of London; Queen Mary University London; Universidade de Sao Paulo
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00927-9
• 发表日期: 2020
• 页码: 185-210
• 关键词: quasi-conformal homeomorphisms; DYNAMICS

摘要

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences Fa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_a$$\end{document} the correspondence Fa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_a$$\end{document} is a mating between a quadratic polynomial Qc(z)=z2+c,c is an element of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_c(z)=z<^>2+c,\,\,c \in \mathbb {R}$$\end{document}, and the modular group Gamma=PSL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma =PSL(2,\mathbb {Z})$$\end{document}. They conjectured that this is the case for every member of the family Fa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_a$$\end{document} which has a in the connectedness locus. We show here that matings between the modular group and rational maps in the parabolic quadratic family Per1(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Per_1(1)$$\end{document} provide a better model: we prove that every member of the family Fa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_a$$\end{document} which has a in the connectedness locus is such a mating.