Equations at Infinity for Critical-Orbit-Relation Families of Rational Maps

Abstract

We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space of degree-d bicritical maps with a marked 4-periodic critical point is a d 2-punctured Riemann surface of genus . We also recover a result of Canci and Vishkautsan, showing that the parameter space of degree-2 rational maps with a marked 5-periodic critical point is a 10-punctured elliptic curve, and identifying its isomorphism class over . We carry out an experimental study of the interaction between dynamically defined points of (such as PCF points or punctures) and the group structure of the underlying elliptic curve.