Hyperbolic Equivariants of Rational Maps

Abstract

Abstract Let $K$ denote either ${\mathbb{R}}$ or ${\mathbb{C}}$. In this article, we study two new equivariants and a new invariant attached to a rational map $f\in K(z)$ under the action of conjugation by ${\operatorname{SL}}_2(K)$. The 1st two equivariants naturally live on real hyperbolic $[K:{\mathbb{Q}}]+1$ space and carry information about the action of $f$ on ${\mathbb{P}}^1(K)$. When $K={\mathbb{C}}$, we study how these objects behave as the map $f$ is iterated; the limits that arise are related to Douady and Earle’s construction of conformal barycenters of measures on $S^2$. We end by giving a description of these objects for maps of degree $d=1$.