Multiplier Spectra and the Moduli Space of Degree 3 Morphisms on P1

TL;DR

This paper investigates the relationship between multiplier spectra and the moduli space of degree 3 rational morphisms on the projective line, extending known results from degree 2 to degree 3 and polynomial cases.

Contribution

It establishes new results on the correspondence between multipliers and moduli space classes for degree 3 rational maps, expanding the understanding beyond degree 2.

Findings

For degree 3 rational maps, the multiplier spectrum determines the conjugacy class.

The paper extends the isomorphism result from degree 2 to degree 3 in specific cases.

New techniques are developed to analyze the moduli space structure for degree 3 maps.

Abstract

The moduli space of degree $d$ morphisms on $\mathbb{P}^1$ has received much study. McMullen showed that, except for certain families of Latt\`es maps, there is a finite-to-one correspondence (over $\mathbb{C}$) between classes of morphisms in the moduli space and the multipliers of the periodic points. For degree 2 morphisms Milnor (over $\mathbb{C}$) and Silverman (over $\mathbb{Z}$) showed that the correspondence is an isomorphism. In this article we address two cases: polynomial maps of any degree and rational maps of degree 3.