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 the topological and combinatorial structures of Lyubich and Minsky’s affine and hyperbolic33-laminations associated with hyperbolic and parabolic quadratic maps.We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same33-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 quotient33-laminations of Douady’s rabbit, the Cauliflower, andz↦z2z \mapsto z^2are described.The descriptions employ a new method oftessellationinside the filled Julia set introduced in Part I [Ergodic Theory Dynam. Systems29(2009), no. 2] that works like external rays outside the Julia set.