Tessellation and Lyubich–Minsky laminations associated with quadratic maps, I: pinching semiconjugacies

Abstract

AbstractWe constructtessellationsof the filled Julia sets of hyperbolic and parabolic quadratic maps. The dynamics inside the Julia sets are then organized by tiles which play the role of the external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperbolic-to-parabolic degenerationswithoutusing quasiconformal deformation. Instead, we achieve this via tessellation and investigation of the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia set.