Dynamical Systems of Correspondences on the Projective Line II: Degrees of Multiplier Maps

TL;DR

This paper investigates the inverse problem of counting isospectral correspondences with identical multipliers on the projective line, providing explicit bounds and correcting previous results using computational and invariant theory methods.

Contribution

It offers a primitive explicit upper bound on the number of rational maps sharing the same multipliers and corrects a prior result on cubic morphisms using two different proof techniques.

Findings

At most O(d^{10d}) rational maps share the same multipliers for fixed and 3-periodic points.

Provided two proofs for a correction to Hutz-Tepper's result on cubic morphisms.

Established bounds and clarified the relationship between multipliers and morphism uniqueness.

Abstract

This paper is a sequel of arXiv:2109.06394. In this paper, we consider a kind of inverse problem of multipliers. The problem is to count number of isospectral correspondences, correspondences which has the same combination of multipliers. We give a primitive explicit upper bound. In particular, for a generic rational map of degree $d$, there are at most $O(d^{10d})$ rational maps with the same combination of multipliers for the fixed points and the 3-periodic points. This paper also includes two proofs of a correction in the errata of a Hutz-Tepper's result, which states that the multipliers of the fixed and 2-periodic points determines generic cubic morphism uniquely. One is done by proceeding the computation in Hutz-Tepper's proof. The other is done by more explicit computation with the help of invariant theory.