Hyperbolic Equivariants of 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2018
- DOI:
- 10.48550/arXiv.1803.07460
- arXiv:
- arXiv:1803.07460
- Bibcode:
- 2018arXiv180307460J
- Keywords:
-
- Mathematics - Dynamical Systems;
- Mathematics - Number Theory
- E-Print:
- 36 pages