Jordan mating is always possible for polynomials

TL;DR

This paper proves that for certain pairs of post-critically finite polynomials with shared super-attracting fixed points, it is always possible to construct a rational map of a specific degree by gluing them along their basins, enabling the creation of rational maps with complex dynamics.

Contribution

The paper introduces a method to construct rational maps by gluing two polynomials along their basins, generalizing previous results and providing a new tool for dynamical systems research.

Findings

Construction of rational maps with prescribed properties

Explicit degree formula for the resulting rational map

Application to generating maps with interesting dynamics

Abstract

Suppose $f$ and $g$ are two post-critically finite polynomials of degree $d_1$ and $d_2$ respectively and suppose both of them have a finite super-attracting fixed point of degree $d_0$. We prove that one can always construct a rational map $R$ of degree $$D = d_1 + d_2 - d_0$$ by gluing $f$ and $g$ along the Jordan curve boundaries of the immediate super-attracting basins. The result can be used to construct many rational maps with interesting dynamics.