Tessellation and Lyubich-Minsky laminations associated with quadratic maps II: Topological structures of 3-laminations

Abstract

According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For example, the differences between the structures of the quotient 3-laminations of Douady's rabbit, the Cauliflower, and $z \mapsto z^2$ are described. The descriptions employ a new method of tessellation inside the filled Julia set introduced in Part I that works like external rays outside the Julia set.