Classification of special curves in the space of cubic polynomials

TL;DR

This paper classifies all special algebraic curves in the parameter space of complex cubic polynomials that contain infinitely many post-critically finite polynomials, confirming a conjecture by Baker and DeMarco.

Contribution

It provides a complete description of special curves in the cubic polynomial parameter space, solving a significant conjecture in complex dynamics.

Findings

All special curves contain infinitely many post-critically finite polynomials.

Speciality of curves with given orbit period and multiplier depends solely on the multiplier being zero.

Confirmed the Baker-DeMarco conjecture for cubic polynomials.

Abstract

We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker and DeMarco for cubic polynomials. We also prove that an irreducible component of the algebraic curve consisting of those cubic polynomials that admit an orbit of any given period and given multiplier is special if and only if the multiplier is 0.