Generic reconstruction of rational maps from multipliers of periods one and two

Abstract

For every integer $d\geq2$ and every field of characteristic different from $2$, we prove that on the moduli space $\mathcal{M}_d$ of degree-$d$ rational maps, the multiplier spectrum morphism formed from the periodic points of periods one and two is birational to the closure of its image. Consequently, over every algebraically closed field of characteristic different from $2$, it is generically injective, which proves a recent conjecture of Ji and Xie in characteristic zero. The proof uses a fixed-index normal form, a non-archimedean degeneration of two-cycles, and birational reconstruction of an affine fixed-point configuration from pair invariants.