Barycentric extension and degenerating sequences of rational maps

We start by investigating the well-known Douady-Earle/Barycentric extension of maps on $S^n$ introduced by Douady and Earle in the 1980s. We prove a regularity result of the extension of its Lipschitz constant. The regularity result allows us to construct a geometric limit of dynamics on an $\mathbf R$-tree for the extension of a degenerating sequence of rational maps. The dynamics on $\mathbf R$-trees gives a natural compactification of marked hyperbolic components. We will compare this compactification with Thurston’s compactification of Teichmüller spaces.