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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.

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
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