Rational maps with Cantor bubble Julia sets

TL;DR

This paper establishes criteria for the existence of Cantor bubble Julia sets in rational maps, constructs various examples, and explores their geometric properties and equivalences.

Contribution

It provides new criteria for Cantor bubble Julia sets in rational maps and constructs examples with specific dynamical and geometric features.

Findings

Criteria for Cantor bubble Julia sets in rational maps

Construction of examples with high-periodic attracting cycles

Conditions for quasisymmetric equivalence to Cantor round bubbles

Abstract

It has been shown that Cantor bubble Julia sets can appear in the dynamics of polynomials and their singular perturbations. In this paper, we present a criterion that guarantees the existence of Cantor bubble Julia sets for certain rational maps with attracting or parabolic fixed points. Moreover, we construct other Cantor bubble Julia sets, including those with high-periodic attracting cycles and those with Hausdorff dimension two. Finally, we give a sufficient condition for Cantor bubble Julia sets to be quasisymmetrically equivalent to Cantor round bubbles.