On the compactness of Julia sets of \(p\)-adic polynomials (Q1434189)

It is easy to see that the Julia set of a polynomial dynamical system over \(\mathbb C\) is a compact subset of \(\mathbb P^1(\mathbb C)\). If one considers \(\mathbb C_p\) (the completion of an algebraic closure of the field of \(p\)-adic numbers) instead of \(\mathbb C\), that is not necessarily so. The author shows that the compactness assumption imposes a strong restriction upon the dynamical system. In particular, if \(J(P_0)\) is the Julia set of a polynomial \(P_0\) of a degree \(\geq 2\), and \(J(P_0)\) is nonempty and compact, then all periodic points are repelling. In this case, the mapping \(P\mapsto J(P)\) is continuous at \(P_0\) with respect to the Hausdorff distance on the space of nonempty bounded closed subsets of \(\mathbb P^1(\mathbb C_p)\).