Points périodiques des fonctions rationnelles dans l'espace hyperbolique $p$-adique

Abstract

We study the dynamics of rational maps with coefficients in the field ${\mathbb{C}}_p$ acting on the hyperbolic space ${\mathbb{H}}_p$. Our main result is that the number of periodic points in ${\mathbb{H}}_p$ of such a rational map is either $0$, $1$ or $\infty$, and we characterize those rational maps having precisely $0$ or $1$ periodic points. The main property we obtain is a criterion for the existence of infinitely many periodic points (of a special kind) in hyperbolic space. The proof of this criterion is analogous to G. Julia's proof of the density of repelling periodic points in the Julia set of a complex rational map.