Theorie ergodique des fractions rationnelles sur un corps ultrametrique

Details Theorie ergodique des fractions rationnelles sur un corps ultrametrique Theorie ergodique des fractions rationnelles sur un corps ultrametrique

2007-09-02

arxiv.org/abs/0709.0092v3

Mathematics

Dynamical Systems

43 pages, 4 figures. In french. Final version. To appear in the Proceedings of the London Math. Soc

Scientific paper

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.

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