Quantitative approximations of the Lyapunov exponent of a rational function over valued fields

TL;DR

This paper provides a quantitative formula to approximate the Lyapunov exponent of rational functions over valued fields, linking it to periodic point multipliers, extending previous convergence results to non-archimedean settings.

Contribution

It introduces a new approximation formula for Lyapunov exponents over valued fields, generalizing earlier convergence results to non-archimedean and archimedean cases.

Findings

Established a quantitative approximation formula for Lyapunov exponents.

Extended convergence results to non-archimedean fields.

Unified approximation approach over various valued fields.

Abstract

We establish a quantitative approximation formula of the Lyapunov exponent of a rational function of degree more than one over an algebraically closed field of characteristic $0$ that is complete with respect to a non-trivial and possibly non-archimedean absolute value, in terms of the multipliers of periodic points of the rational function. This quantifies both our former convergence result over general fields and the one-dimensional version of Berteloot--Dupont--Molino's one over archimedean fields.