Julia sets as buried Julia components

TL;DR

This paper proves conditions under which a rational map can be constructed to contain a buried Julia component with dynamics conjugate to a given map, and provides degree bounds and explicit examples.

Contribution

It establishes necessary and sufficient conditions for the existence of buried Julia components with conjugate dynamics and constructs explicit examples with degree bounds.

Findings

Existence of buried Julia components under specific conditions.

Degree bounds for constructed rational maps.

Explicit examples of rational maps with buried Julia curves.

Abstract

Let $f$ be a rational map with degree $d\geq 2$ whose Julia set is connected but not equal to the whole Riemann sphere. It is proved that there exists a rational map $g$ such that $g$ contains a buried Julia component on which the dynamics is quasiconformally conjugate to that of $f$ on the Julia set if and only if $f$ does not have parabolic basins and Siegel disks. If such $g$ exists, then the degree can be chosen such that $\text{deg}(g)\leq 7d-2$. In particular, if $f$ is a polynomial, then $g$ can be chosen such that $\text{deg}(g)\leq 4d+4$. Moreover, some quartic and cubic rational maps whose Julia sets contain buried Jordan curves are also constructed.