Hyperbolic Equivariants of Rational Maps

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