Checkerboard Julia Sets for Rational Maps

TL;DR

This paper studies the Julia sets of a family of rational maps, showing they are always homeomorphic and classifying conjugacy classes based on a new invariant called the minimal rotation number.

Contribution

It introduces the concept of the minimal rotation number to classify conjugacy classes of Julia sets for these rational maps.

Findings

Julia sets are always homeomorphic within the family.

Conjugacy classes are characterized by parameters satisfying specific algebraic relations.

An exact count of conjugacy classes in the main cardioids is provided.

Abstract

In this paper, we consider the family of rational maps $$\F(z) = z^n + \frac{\la}{z^d},$$ where $n \geq 2$, $d\geq 1$, and$\la \in \bbC$. We consider the case where $\la$ lies in the main cardioid of one of the $n-1$ principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps $\F$ and $F_\mu$ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy $\mu = \nu^{j(d+1)}\la$ or $\mu = \nu^{j(d+1)}\bar{\la}$ where $j \in \bbZ$ and $\nu$ is an $n-1^{\rm st}$ root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.