Approximation of non-archimedean Lyapunov exponents and applications over global fields

TL;DR

This paper develops a uniform approximation formula for non-archimedean Lyapunov exponents of rational maps over global fields, with applications to height comparisons, multiplier map finiteness, and characterizations of isotrivial maps.

Contribution

It introduces a new non-archimedean approximation method for Lyapunov exponents with explicit error control, extending previous archimedean results and enabling several applications.

Findings

Established a uniform approximation formula for Lyapunov exponents over non-archimedean fields.

Provided bounds on Lyapunov exponent blow-up near poles in rational map families.

Connected Lyapunov exponent approximation to height comparisons and multiplier map finiteness.

Abstract

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show: - a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space $\mathcal{M}_d(\bar{\mathbb{Q}})$, - two improvements of McMullen's finiteness of the multiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period, - a characterization of non-affine isotrivial rational maps defined over the function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.