Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies

TL;DR

This paper introduces tessellations of filled Julia sets for quadratic maps and constructs pinching semiconjugacies during hyperbolic-to-parabolic degenerations without quasiconformal methods.

Contribution

It develops a novel tessellation approach to organize Julia set dynamics and constructs continuous pinching semiconjugacies using internal coordinate analysis.

Findings

Tessellations effectively organize Julia set dynamics.

Pinching semiconjugacies are constructed without quasiconformal deformation.

The approach applies to hyperbolic and parabolic quadratic maps.

Abstract

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 pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and investigation on the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia sets.