Irreducibility of the set of cubic polynomials with one periodic critical point

TL;DR

This paper proves that the algebraic sets formed by cubic polynomials with a critical point of a fixed period are irreducible, answering a question posed by Milnor about the structure of these sets.

Contribution

It establishes the irreducibility of the loci of cubic polynomials with a marked critical point of a given period, advancing understanding of polynomial dynamics.

Findings

The set Sn is irreducible for all n>0.

The space of cubic polynomials with a marked critical point is isomorphic to C^2.

The result confirms a conjecture about the algebraic structure of these dynamical loci.

Abstract

The space of monic centered cubic polynomials with marked critical points is isomorphic to C^2. For each n>0, the locus Sn formed by all polynomials with a specified critical point periodic of exact period n forms an affine algebraic set. We prove that Sn is irreducible, thus giving an affirmative answer to a question posed by Milnor. (This manuscript has been withdrawn)