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 a 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 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 zero characteristic case.