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

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinch-ing semiconjugacies associated with hyperbolic-to-parabolic degenerations without us-ing quasiconformal deformation. Instead we use tessellation and investigation on the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia set.