Limits of rational maps, R-trees and barycentric extension

Abstract

In this paper, we show that one can naturally associate a limiting dynamical system $F:T⟶T$ on an $\mathbb{R}$-tree to any degenerating sequence of rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The construction of $F$ is in $2$ steps: first we use barycentric extension to get $\E f_n : \Hyp^3 \longrightarrow \Hyp^3$; second, we take appropriate limit on rescalings of hyperbolic space. An important ingredient we prove here is that the Lipschitz constant depends only on the degree of the rational map. We show that the dynamics of $F$ records the limiting length spectra of the sequence $f_n$.

41 pages, 5 figure. Final version

References in corpus (3)

• Conformally Natural extensions revisited
• On hyperbolic rational maps with finitely connected Fatou sets
• Trees, length spectra for rational maps via barycentric extensions and Berkovich spaces

Cited by in corpus (2)

• On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps
• Combination Theorems in Groups, Geometry and Dynamics