Constructing rational maps with cluster points using the mating operation

TL;DR

This paper demonstrates how all admissible rational maps with specific cluster cycles can be constructed through polynomial matings, providing a combinatorial classification and analyzing shared matings.

Contribution

It establishes a method to construct rational maps with cluster points via mating of polynomials and classifies the polynomials involved.

Findings

All admissible rational maps with fixed or period two cluster cycles can be constructed by mating polynomials.

In the one cluster case, the polynomial must be an n-rabbit.

In the two cluster case, the polynomial is either a double rabbit or a secondary map in the wake of the double rabbit.

Abstract

In this article, we show that all admissible rational maps with fixed or period two cluster cycles can be constructed by the mating of polynomials. We also investigate the polynomials which make up the matings that construct these rational maps. In the one cluster case, one of the polynomials must be an $n$-rabbit and in the two cluster case, one of the maps must be either $f$, a "double rabbit", or $g$, a secondary map which lies in the wake of the double rabbit $f$. There is also a very simple combinatorial way of classifiying the maps which must partner the aforementioned polynomials to create rational maps with cluster cycles. Finally, we also investigate the multiplicities of the shared matings arising from the matings in the paper.