Non-Archimedean techniques and dynamical degenerations

TL;DR

This paper introduces non-Archimedean methods using Berkovich spaces to study rational map degenerations, providing new tools for analyzing measure convergence and Lyapunov exponents.

Contribution

It offers an alternative to Luo's hyperbolic ultra-limit approach by developing hybrid spaces with Berkovich theory for dynamical analysis.

Findings

Proves convergence of equilibrium measures in degenerating rational maps.

Determines asymptotics of Lyapunov exponents during degeneration.

Provides a new non-Archimedean framework for complex dynamics.

Abstract

We develop non-Archimedean techniques to analyze the degeneration of a sequence of rational maps of the complex projective line. We provide an alternative to Luo's method which was based on ultra-limits of the hyperbolic 3-space. We build hybrid spaces using Berkovich theory which enable us to prove the convergence of equilibrium measures, and to determine the asymptotics of Lyapunov exponents.