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

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We study the dynamics of rational maps with coefficients in the field {\Bbb C}_p acting on the hyperbolic space {\Bbb H}_p . Our main result is that the number of periodic points in {\Bbb 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.