Hyperbolic Equivariants of Rational Maps. (30th September 2021)

• Record Type: Journal Article
• Authors: Jacobs, Kenneth
• Is Part Of: International mathematics research notices. Volume 2023:Number 1(2023)
• Page Start: 1
• Page End: 53
• Publication Date: 2021-09-30
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab266
• Languages: English
• ISSNs: 1073-7928

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 $d=1$ .