Hyperbolic Equivariants of Rational Maps

2018/03/20 by Jacobs, Kenneth
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1803.07460

Abstract

Let K denote either ℝ or ℂ. In this article, we introduce two new equivariants associated to a rational map f∈ K(z). These objects naturally live on a real hyperbolic space, and carry information about the action of f on ℙ¹(K). When K=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.

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