Equations at infinity for critical-orbit-relation families of rational maps

TL;DR

This paper develops new techniques using compactifications of Hurwitz spaces to analyze parameter spaces of rational maps with critical orbit relations, revealing their geometric structures and special points.

Contribution

It introduces methods to study families of rational maps via compactified Hurwitz spaces and characterizes specific parameter spaces as punctured Riemann surfaces and elliptic curves.

Findings

Parameter space of degree-d bicritical maps with a 4-periodic critical point is a d^2-punctured Riemann surface.

Parameter space of degree-2 maps with a 5-periodic critical point is a 10-punctured elliptic curve.

Experimental analysis of special points and group structure interactions on the elliptic curve.

Abstract

We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps $\mathbb{P}^1\to\mathbb{P}^1$ defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space $\mathrm{Per}_{d,n}$ of degree-$d$ bicritical maps with a marked 4-periodic critical point is a $d^2$-punctured Riemann surface of genus $\frac{(d-1)(d-2)}{2}$. We also show that the parameter space $\mathrm{Per}_{2,5}$ of degree-2 rational maps with a marked 5-periodic critical point is a 10-punctured elliptic curve, and we identify its isomorphism class over $\mathbb{Q}$. We carry out an experimental study of the interaction between dynamically defined points of $\mathrm{Per}_{2,5}$ (such as PCF points or punctures) and the group structure of the underlying elliptic curve.