Closure of Periodic Points Over a Non-Archimedean Field

Abstract

The closure of the periodic points of rational maps over a non‐archimedean field is studied. An analogue of Montel's theorem over non‐archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non‐archimedean field is contained in the closure of the periodic points.

Book Review: The arithmetic of dynamical systems

“…We can therefore consider the action of φ on P 1 (C p ), the p-adic analogue of the Riemann sphere. Dynamics over non-archimedean fields is a newer research area with fewer expositions, although some early research papers such as [2,3,5,13] provide something of an introduction to the subject. Nevertheless, much of the same theory of Fatou and Julia sets can be carried out in the non-archimedean context; the denominators of 12 in Example 1.2, for instance, can be understood as coming from the 2-adic and 3-adic Julia sets of φ(z) = z 2 − 133/144.…”

Wandering Domains in Non-Archimedean Polynomial Dynamics

“…Moreover, for such x, the component of φ n (x) is precisely φ n (V ). We recall Hsia's criterion [12] for equicontinuity, which is a non-archimedean analogue of the Montel-Carathéodory Theorem.…”

Heights and totally p-adic numbers

“…The Berkovich Julia set In complex dynamics, the Julia set is the closure of the repelling periodic points. In the classic non-archimedean setting this is an open conjecture (see [18]). The next proposition will be useful in the proof of Theorem 1.1.…”