Dynamical systems of correspondences on the projective line I: Moduli spaces and multiplier maps

Abstract

We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We define the moduli space D y n d , e Dyn_{d,e} of degree ( d , e ) (d,e) correspondences. We construct a family ρ c : D y n d , e ⇢ D y n 1 , d + e − 1 \rho _c : Dyn_{d,e} \dashrightarrow Dyn_{1,d+e-1} of rational maps representation-theoretically. Here we note that D y n 1 , d + e − 1 Dyn_{1,d+e-1} is identical to the moduli space of the usual dynamical systems of degree d + e − 1 d+e-1 . We show that the moduli space D y n d , e Dyn_{d,e} is rational by using ρ c \rho _c . Moreover, the multiplier maps for the fixed points factor through ρ c \rho _c . Furthermore, we show the Woods Hole formulae for correspondences of different degrees are also related by ρ c \rho _c and obtain another representation-theoretically simplified form of the formula.