Multiplier spectra and the moduli space of degree 3 morphisms on ℙ¹

The moduli space of degree d morphisms on ℙ1 has received much study. McMullen showed that, except for certain families of Lattès maps, there is a finite-to-one correspondence (over ℂ) between classes of morphisms in the moduli space and the multipliers of the periodic points. For degree 2 morphisms, Milnor (over ℂ) and Silverman (over ℤ) showed that the correspondence is an isomorphism [8, 10]. In this article, we address two cases with algebraic methods: polynomial maps of any degree and rational maps of degree 3.