Special curves and postcritically-finite polynomials

TL;DR

This paper characterizes when certain rational curves in the moduli space of complex polynomials contain infinitely many postcritically-finite maps, using a mix of number theory and complex analysis, and proposes a related density conjecture.

Contribution

It provides a complete characterization of rational curves with infinitely many PCF maps and introduces a conjecture on the Zariski density of PCF maps in rational map subvarieties.

Findings

Infinitely many PCF maps occur on certain polynomial parameterized curves with exactly one active critical point.

The curve $ ext{Per}_1( extstyle{rac{ ext{constant}}{ ext{degree}}})$ contains infinitely many PCF maps only when the parameter is zero.

The proofs combine arithmetic equidistribution and univalent function theory techniques.

Abstract

We study the postcritically-finite (PCF) maps in the moduli space of complex polynomials $\mathrm{MP}_d$. For a certain class of rational curves $C$ in $\mathrm{MP}_d$, we characterize the condition that $C$ contains infinitely many PCF maps. In particular, we show that if $C$ is parameterized by polynomials, then there are infinitely many PCF maps in $C$ if and only if there is exactly one active critical point along $C$, up to symmetries; we provide the critical orbit relation satisfied by any pair of active critical points. For the curves $\mathrm{Per}_1(\lambda)$ in the space of cubic polynomials, introduced by Milnor (1992), we show that $\mathrm{Per}_1(\lambda)$ contains infinitely many PCF maps if and only if $\lambda=0$. The proofs involve a combination of number-theoretic methods (specifically, arithmetic equidistribution) and complex-analytic techniques (specifically, univalent function theory). We provide a conjecture about Zariski density of PCF maps in subvarieties of the space of rational maps, in analogy with the Andr\'e-Oort Conjecture from arithmetic geometry.