Dynamical Systems of Correspondences on the Projective Line I: Moduli Spaces and Multiplier Maps

TL;DR

This paper studies moduli spaces of dynamical correspondences on the projective line, proving their rationality and analyzing multiplier maps, thereby extending classical dynamical systems theory to correspondences.

Contribution

It introduces a framework for moduli spaces of correspondences, proves their rationality, and explicitly describes the reduced multiplier maps using representation theory.

Findings

Proved the rationality of moduli spaces of correspondences.

Established the reduction of multiplier maps via projection.

Derived explicit forms of the multiplier index theorem for correspondences.

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 obtain the rationality of the moduli spaces. The rationality of the moduli space of degree $(d,e)$ correspondences is obtained from a representation-theoretic projection to the one for the usual dynamical systems of degree $d+e-1$. We also show that the multiplier maps for the fixed points and the multiplier index theorem (Woods Hole formula) are also reduced through the projection and obtain the reduced form explicitly.