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

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. 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 structure of the quotient 3-lamination of Douady’s rabbit is given by pinching and plumping with “1/3-Dehn twist” of the lower end of quotient 3-lamination of z 7→ z2. The descriptions employ a new method of tessellation inside the filled Julia set introduced in Part I [Ka3] that works like external rays outside the Julia set.