We study the dynamics of rational maps with coefficients in the field acting on the hyperbolic space . Our main result is that the number of periodic points in of such a rational map is either , or , and we characterize those rational maps having precisely or 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.
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Juan Rivera-Letelier, Points périodiques des fonctions rationnelles dans l'espace hyperbolique -adique. Comment. Math. Helv. 80 (2005), no. 3, pp. 593–629DOI 10.4171/CMH/27