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Dynamics of rational functions over local fields. (Dynamique des fonctions rationnelles sur des corps locaux.) (French) Zbl 1140.37336
de Melo, Welington (ed.) et al., Geometric methods in dynamics (II). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems held at IMPA, Rio de Janeiro, Brazil, July 2000, to celebrate Jacob Palis’ 60th birthday. Paris: Société Mathématique de France (ISBN 2-85629-139-2/pbk). Astérisque 287, 147-230 (2003).
Summary: Let \(p > 1\) be a prime number, \(\mathbb Q_p\) the field of \(p\)-adic numbers and let \(\mathbb C_p\) be the smallest complete extension of \(\mathbb Q_p\) that is algebraically close. This work is dedicated to the study of the dynamics of rational functions on the projective line \({\mathbb P}(\mathbb C_p)\). To each rational function \(R \in \mathbb C_p(z)\) we associate its quasi-periodicity domain, which is equal to the interior of the set of points in \({\mathbb P}(\mathbb C_p)\) that are recurrent by \(R\). We give several characterizations of the quasi-periodicity domain and we describe its local and global dynamics. Like in the complex case there is a partition of the line \({\mathbb P}(\mathbb C_p)\) in the Fatou set and the Julia set. By analogy to the complex case we make the following non-wandering conjecture: every wandering disc is attracted to an attracting cycle. We prove that this holds if and only if every point in the Fatou set is either attracted to an attracting cycle or if it is mapped to the quasi-periodicity domain under forward iteration. We prove that analytic components of the domain of quasi-periodicity (which are the \(p\)-adic analogues of Siegel discs and Herman rings) are open affinoïdes (that is, they have simple geometry) and we describe the dynamics in a given component.
For the entire collection see [Zbl 1031.37003].

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
14G20 Local ground fields in algebraic geometry
39B12 Iteration theory, iterative and composite equations