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Degenerations of complex dynamical systems. II: Analytic and algebraic stability. (English) Zbl 1343.37089
Summary: We study pairs \((f, \Gamma )\) consisting of a non-Archimedean rational function \(f\) and a finite set of vertices \(\Gamma\) in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set \(\Gamma\). As a byproduct, we deduce that meromorphic maps preserving the fibers of a rationally-fibered complex surface are algebraically stable after a proper modification. The first article in this series [the authors, Forum Math. Sigma 2, Article ID e6, 36 p. (2014; Zbl 1308.37023)] examined the limit of the equilibrium measures for a degenerating 1-parameter family of rational functions on the Riemann sphere. Here we construct a convergent countable-state Markov chain that computes the limit measure. A classification of the periodic Fatou components for non-Archimedean rational functions, due to J. Rivera-Letelier [in: Geometric methods in dynamics (II). Volume in honor of Jacob Palis. In part papers presented at the international conference on dynamical systems. Paris: Société Mathématique de France. 147–230 (2003; Zbl 1140.37336)], plays a key role in the proofs of our main theorems. The appendix contains a proof of this classification for all tame rational functions.

MSC:
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P50 Dynamical systems on Berkovich spaces
37P20 Dynamical systems over non-Archimedean local ground fields
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