On the periodic points of rational maps in ultrametric dynamics (Q2759145)

It is well known that the Julia set of a complex rational function of degree \(\geq 2\) coincides with the closure of the set of all its repulsive periodic points. \textit{L. Hsia} [Compos. Math. 100, 277--304 (1996; Zbl 0851.14001)] asked whether the same is true for rational mappings over \(\mathbb{C}_p\), the completion of an algebraic closure of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. The author shows that this is indeed so in the case of rational functions having at least one repulsive periodic point (it is not clear at this moment, whether in the remaining case the Julia set is empty). The proof is based on an extension of an equicontinuity criterion of \textit{L. Hsia} [J. Lond. Math. Soc. (2) 62, 685--700 (2000; Zbl 1022.11060)].