Dynamical systems with a parallel tensor

TL;DR

This paper classifies certain rational maps on the projective line that preserve a special parallel tensor, linking them to maps that maintain a parabolic orbifold structure.

Contribution

It provides a classification of rational maps with a parallel contravariant tensor, revealing their connection to parabolic orbifold preservation.

Findings

Maps preserve a parabolic orbifold structure.

Existence of a parallel contravariant tensor characterizes these maps.

Classification extends understanding of rational maps with special geometric properties.

Abstract

We classify those rational maps $f: \mathbb{P}^1 \to \mathbb{P}^1$ for which there exists a contravariant tensor $q$ which is parallel, i.e. such that $f^*q // q$, by proving that such maps preserve a parabolic orbifold.