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

2020/08/23 by Rohini Ramadas, Ramadas, Rohini, Rob Silversmith +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2008.10095

openalex publication_date 2020/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps ℙ¹→ℙ¹ defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space Per_d,n of degree-d bicritical maps with a marked 4-periodic critical point is a d²-punctured Riemann surface of genus ((d-1)(d-2))/(2). We also show that the parameter space 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 ℚ. We carry out an experimental study of the interaction between dynamically defined points of Per_2,5 (such as PCF points or punctures) and the group structure of the underlying elliptic curve.

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