Theorie ergodique des fractions rationnelles sur un corps ultrametrique

TL;DR

This paper explores the ergodic properties of rational maps over non-archimedean fields, constructing invariant measures and analyzing their mixing, entropy, and reduction properties.

Contribution

It introduces the first invariant measure for such maps and studies its ergodic and entropy characteristics, linking entropy to potential good reduction.

Findings

Constructed a natural invariant measure for rational maps over non-archimedean fields.

Proved the measure is exponentially mixing and satisfies the central limit theorem.

Established bounds on metric and topological entropy, and related entropy to potential good reduction.

Abstract

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure m_R which reprensents the asymptotic distribution of preimages of non-exceptional point. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of m_R, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.