Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a Levy cycle

TL;DR

This paper presents a detailed example of non-mateable degree 3 polynomials without Levy cycles using slow polynomial mating, with visualizations and conjectures about their Julia sets and related rational maps.

Contribution

It provides the first conformally correct pictures of such non-mateable polynomials and explores their limit Julia sets, connecting them to degree 6 rational maps.

Findings

Visualizations of slow polynomial mating for degree 3 polynomials

Identification of Julia set limits related to degree 6 rational maps

Conjectural interpretation involving pinched spheres

Abstract

After giving an introduction to the procedure dubbed slow polynomial mating and stating a conjecture relating this to other notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree 3 post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree 6 rational map. We give a conjectural interpretation of this in terms of pinched spheres and show further conformal representations.