Trees, Berkovich Spaces and the Barycentric Extension in Complex Dynamics

Abstract

A metric space T is called an R-tree if any two points x, y ∈ T can be connected by a unique topological arc [x, y] ⊂ T which is isometric to an interval in R. R-trees are natural generalizations to finite trees and simplicial trees, and have many applications in mapping class groups, Teichmüller theory, hyperbolic 3-manifold and Kleinian groups and etc. In this work, we will give a new construction of R-trees in complex dynamics using barycentric extensions. We will establish the relation between the barycentric construction and the Berkovich construction via the complexified Robison’s field. As an application, we will also use our construction to classify all hyperbolic components that admits degenerating sequences with bounded multipliers.