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

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