Irreducibility of periodic curves in cubic polynomial moduli space

TL;DR

This paper proves that in the moduli space of complex cubic polynomials with a marked critical point, the set of polynomials with a critical point of a fixed period forms an irreducible curve, resolving a long-standing question.

Contribution

The authors establish the irreducibility of loci of cubic polynomials with a marked critical point of fixed period, answering Milnor's question from the 1990s.

Findings

Loci of polynomials with a periodic critical point are irreducible curves.

The result applies to all periods p ≥ 1.

Provides a complete answer to Milnor's question on this topic.

Abstract

In the moduli space of complex cubic polynomials with a marked critical point, given any p>=1, we prove that the loci formed by polynomials with the marked critical point periodic of period p is an irreducible curve. Thus answering a question posed by Milnor in the 90's.