A family of rational maps with buried Julia components

TL;DR

This paper introduces a new family of hyperbolic rational maps with disconnected Julia sets containing various types of buried Julia components, including points, Jordan curves, and more complex structures, some of degree 3.

Contribution

It presents the first explicit examples of rational maps with buried Julia components beyond points and Jordan curves, including maps of degree 3.

Findings

Existence of buried Julia components of multiple types

Construction of rational maps with degree 3 with explicit formulas

Encoding dynamics via weighted dynamical trees

Abstract

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least 5. This paper introduce a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover this family contains some rational maps of degree 3 with explicit formula that answers a question McMullen raised.