An Algebraic Proof of Thurston's Rigidity for a Polynomial

TL;DR

This paper provides an algebraic proof of Thurston's rigidity theorem for certain polynomial maps with finite critical orbits, demonstrating finiteness outside a specific family and presenting counterexamples in the wildly ramified case.

Contribution

It offers the first algebraic proof of Thurston's rigidity for tamely ramified maps with at least one periodic critical point, expanding understanding of polynomial dynamics.

Findings

Finiteness of such maps outside a well-understood family

Construction of counterexamples in wildly ramified cases

Algebraic methods applied to Thurston's rigidity theorem

Abstract

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of fixed degree and critical orbit length. We provide an algebraic proof of this fact for tamely ramified maps for which at least one of the critical points is periodic. We also produce wildly ramified counterexamples.