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

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.