# zbMATH — the first resource for mathematics

Periodic points of rational functions in the $$p$$-adic hyperbolic space. (Points périodiques des fonctions rationnelles dans l’espace hyperbolique $$p$$-adique.) (French) Zbl 1140.37337
Summary: 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.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 37E99 Low-dimensional dynamical systems 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
Full Text: