Continuity of Julia sets and invariant rays

TL;DR

This paper explores how Julia sets and invariant rays behave under perturbations of rational maps with parabolic cycles, linking geometric convergence with multiplier convergence through parabolic implosion theory.

Contribution

It establishes new equivalences between Julia set convergence, invariant rays, and multiplier convergence using parabolic implosion and edge dynamics analysis.

Findings

Julia sets converge in Hausdorff sense under perturbations

Invariant rays exhibit continuity properties during perturbations

Multiplier convergence correlates with Julia set stability

Abstract

For certain typical perturbations $(f_n)_n$ of a rational map $f$ with parabolic cycles, we investigate the relations between the Hausdorff convergence of Julia sets and invariant rays, and the horocyclic convergence of multipliers of periodic points. We establish several equivalent characterizations by means of parabolic implosion theory. This builds upon an analysis of the edge dynamics on the tree for the gate structure induced by the perturbation. The edge dynamics which are driven by Oudkerk's algorithm, are used to trace the orbits for the near parabolic perturbations.