The McMullen Map in Positive Characteristic

TL;DR

This paper extends McMullen's parametrization of rational maps by multipliers from complex to positive characteristic fields, showing that generically maps are determined by their multipliers with finitely many exceptions, and applies this to root-finding algorithms.

Contribution

It generalizes McMullen's method to positive characteristic using rigid analysis, establishing a finite-to-one parametrization of rational maps by multipliers over finite fields.

Findings

Rational maps are generically parametrized by multipliers in positive characteristic.

The set of exceptions corresponds to finitely many multiplier spectra.

No convergent purely iterative root-finding algorithm exists over certain non-archimedean fields.

Abstract

McMullen proved the moduli space of complex rational maps can be parametrized by the spectrum of all periodic-point multipliers up to a finite amount of data, with the well-understood exception of Latt\`{e}s maps. We generalize his method to large positive characteristic. McMullen's method is analytic; a modified version of the method using rigid analysis works over a function field over a finite field of characteristic larger than the degree of the map. Over a finite field with such characteristic it implies that, generically, rational maps can indeed be parametrized by their multiplier spectra up to a finite-to-one map. Moreover, the set of exceptions, that is positive-dimension varieties in moduli space with identical multipliers, maps to just a finite set of multiplier spectra. We also prove an application, generalizing a result of McMullen over the complex numbers: there is no generally convergent purely iterative root-finding algorithm over a non-archimedean field whose residue characteristic is larger than either the degree of the algorithm or the degree of the polynomial whose roots the algorithm finds.