Hyperbolic Equivariants of Rational Maps

TL;DR

This paper introduces two new hyperbolic space equivariants for rational maps over real or complex fields, linking their asymptotic behavior to entropy measures and extending Rumely's non-Archimedean work.

Contribution

It defines novel equivariants on hyperbolic space for rational maps and explores their properties, including special cases and connections to entropy and non-Archimedean theory.

Findings

Equivariants encode action of rational maps on hyperbolic space

Asymptotics relate to conformal barycenter of entropy measure

Complete description for degree 1 rational maps

Abstract

Let $K$ denote either $\mathbb{R}$ or $\mathbb{C}$. In this article, we introduce two new equivariants associated to a rational map $f\in K(z)$. These objects naturally live on a real hyperbolic space, and carry information about the action of $f$ on $\mathbb{P}^1(K)$. When $K=\mathbb{C}$ we relate the asymptotic behavior of these equivariants to the conformal barycenter of the measure of maximal entropy. We also give a complete description of these objects for rational maps of degree $d=1$. The constructions in this article are based on work of Rumely in the context of rational maps over non-Archimedean fields; similarities between the two theories are highlighted throughout the article.