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

Rivera-Letelier, Juan

Comment. Math. Helv. 80, No. 3, 593-629 (2005).

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

Keywords:

\(p\)-adic fields; rational maps; hyperbolic space; periodic points