Rational functions periodic points in p-adic hyperbolic space

We study the dynamics of rational maps with coefficients in the field C-p acting on the hyperbolic space H-p. Our main result is that the number of periodic points in H-p of such a rational map is either 0, 1 or infinity, 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.