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Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics

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 the variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and Rivera-Letelier, using a dynamical Diophantine approximation theorem by Silverman and by Szpiro--Tucker. We also give a proof of a general equidistribution theorem for possibly moving targets, which is due to Lyubich in the archimedean case and due to Favre and Rivera-Letelier for constant targets in the non-archimedean and any characteristic case and for moving targets in the non-archimedean and 0 characteristic case.