A dynamical system over a non-archimedean field

TL;DR

This paper surveys the theory of dynamical systems over non-archimedean fields, focusing on potential theory, moduli spaces, height functions, and recent advances including McMullen's finiteness theorem and Silverman's conjecture.

Contribution

It provides an expository overview of recent developments in non-archimedean dynamics, including new results on moduli spaces and height functions, and connects complex and non-archimedean perspectives.

Findings

Survey of Rumely's equivariants in non-archimedean dynamics

Precise version of McMullen's finiteness theorem

Effective solution of Silverman's conjecture

Abstract

This is an expository article, originally written in Japanese, on a dynamical system over a non-archimedean field. The main viewpoint is from complex and non-archimedean potential theories. After quickly introducing the Berkovich projective line, the dynamical moduli space as a scheme, and the various height functions on the space of rational functions and on the dynamical moduli space, we first survey our study of Rumely's new equivariants in non-archimedean dynamics and then survey our complex geometric and arithmetic studies of the dynamical moduli space from our joint works with Thomas Gauthier and Gabriel Vigny. The latter include a precise version of McMullen's finiteness theorem on formally exact multiplier spectra and an effective solution of Silverman's conjecture on a comparison between the moduli height and the critical height (qualitatively, the Silverman-Ingram theorem). The final topic is a degeneration of complex dynamics.