The multiplier spectrum morphism is generically injective

TL;DR

This paper proves that the multiplier spectrum morphism for rational maps on the projective line is generically injective, strengthening McMullen's quasi-finiteness result and answering an open question in the field.

Contribution

It establishes the generic injectivity of the multiplier spectrum morphism, advancing understanding of the moduli space of rational maps beyond previous quasi-finiteness results.

Findings

Multiplier spectrum morphism is generically injective

Strengthens McMullen's quasi-finiteness theorem

Answers a question of McMullen and Poonen

Abstract

In this paper, we consider the multiplier spectrum of periodic points, which is a natural morphism defined on the moduli space of rational maps on the projective line. A celebrated theorem of McMullen asserts that aside from the well-understood flexible Latt\`es family, the multiplier spectrum morphism is quasi-finite. In this paper, we strengthen McMullen's theorem by showing that the multiplier spectrum morphism is generically injective. This answers a question of McMullen and Poonen.