Matings of Cubic polynomials with a fixed critical point, Part I: Thurston Obstructions

TL;DR

This paper proves that for certain degree 3 Thurston maps with fixed critical points, any irreducible obstruction must contain a Levy cycle, with implications for matings of postcritically finite cubic polynomials.

Contribution

It establishes that all irreducible obstructions in these maps contain Levy cycles, advancing understanding of Thurston obstructions in cubic polynomial matings.

Findings

Any irreducible obstruction for the specified Thurston maps contains a Levy cycle.

Obstructions to matings of postcritically finite cubic polynomials with fixed critical points also contain Levy cycles.

Examples of the described obstructions are provided in the appendix.

Abstract

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic polynomials each having a fixed critical point, then any obstruction to the mating $f \mate g$ contains a Levy cycle. We end with an appendix to show examples of the obstructions described in the paper.