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Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-Archimedean dynamics. (English) Zbl 1302.37070

Summary: We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on a variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to C. Favre and J. Rivera-Letelier [Proc. Lond. Math. Soc. (3) 100, No. 1, 116–154 (2010; Zbl 1254.37064)], using a dynamical Diophantine approximation theorem by J. H. Silverman [Duke Math. J. 71, No. 3, 793–829 (1993; Zbl 0811.11052)] and by L. Szpiro and T. J. Tucker [in: Number theory, analysis and geometry. Berlin: Springer. 609–638 (2012; Zbl 1283.37075)]. We also give a proof of a general equidistribution theorem for possibly moving targets, which is due to M. Yu. Lyubich [Ergodic Theory Dyn. Syst. 3, 351–385 (1983; Zbl 0537.58035)] in the archimedean case and to Favre and Rivera-Letelier [loc. cit.] for constant targets in the non-archimedean and any characteristic case, and for moving targets in the non-archimedean and zero characteristic case.

MSC:

37P50 Dynamical systems on Berkovich spaces
11S82 Non-Archimedean dynamical systems
37P20 Dynamical systems over non-Archimedean local ground fields
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