Shared matings in $V_2$

TL;DR

This paper introduces a new constructive approach to prove the existence of shared matings in a specific class of rational maps with super-attracting 2-cycles, avoiding the use of Thurston's Theorem.

Contribution

It provides a novel constructive method for establishing shared matings in the class $V_2$, expanding understanding without relying on Thurston's Theorem.

Findings

Established existence of shared matings in $V_2$

Developed a constructive proof method

Extended previous work on shared matings

Abstract

We give a new constructive method to prove existence of shared matings in the special class $V_2$ consisting of rational maps with a super-attracting $2$-cycle (up to M\"obius conjugacy). The proof does not use Thurston's Theorem on branched coverings on the Riemann sphere. The background to this paper is the master thesis of L. Pedersen (Ume\r{a} University, 2014), where one special shared mating was studied.