Théorie ergodique des fractions rationnelles sur un corps ultramétrique

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 ρR which represents the asymptotic distribution of preimages of non-exceptional points. We show that this measure is exponentially mixing, and satisfies the central limit theorem. We prove some general bounds on the metric entropy of ρR, and on the topological entropy of R. We finally prove that rational maps with vanishing topological entropy have potential good reduction.