Trees, length spectra for rational maps via barycentric extensions, and Berkovich spaces

Abstract

In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere $\hat{\mathbb{C}}$ using $\mathbb{R}$-trees. As an analogue of isometric group actions on $\mathbb{R}$-trees for Kleinian groups, we give two constructions for limiting dynamics on $\mathbb{R}$-trees: one geometric and one algebraic. The geometric construction uses the limit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson’s field. We show that the two approaches are equivalent. As an application, we use it to give a classification of hyperbolic components of rational maps that admit degeneracies with bounded multipliers.