  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.
