Rational maps whose Julia sets are Cantor circles

TL;DR

This paper characterizes and constructs rational maps with Julia sets that are Cantor circles, providing explicit examples and showing their topological conjugacy properties, including non-hyperbolic cases.

Contribution

It introduces a family of rational maps with Cantor circle Julia sets and proves their uniqueness up to topological conjugacy, including non-hyperbolic examples.

Findings

Explicit family of rational maps with Cantor circle Julia sets

Uniqueness of such maps up to topological conjugacy

Construction of non-hyperbolic examples

Abstract

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.