Periodic points of rational functions in the \(p\)-adic hyperbolic space (Q2388659)

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.