A rigidity result for some parabolic germs

TL;DR

This paper proves a rigidity theorem for unicritical polynomials with parabolic cycles, showing that conformally conjugate parabolic germs imply the polynomials are affinely conjugate, thus establishing a strong structural uniqueness result.

Contribution

It establishes a rigidity result linking local conformal conjugacy of parabolic germs to global affine conjugacy of unicritical polynomials.

Findings

Conformal conjugacy of parabolic germs implies affine conjugacy of the polynomials.

The result applies specifically to unicritical polynomials with parabolic cycles.

Provides a new understanding of the local-global relationship in polynomial dynamics.

Abstract

The goal of this article is to prove a rigidity result for unicritical polynomials with parabolic cycles. More precisely, we show that if two unicritical polynomials have conformally conjugate parabolic germs, then the polynomials are affinely conjugate.