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

Yûsuke Okuyama¹

¹ Division of Mathematics Kyoto Institute of Technology Sakyo-ku, Kyoto 606-8585, Japan

Acta Arithmetica, Tome 161 (2013) no. 2, pp. 101-125

Résumé

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.

Détail

DOI : 10.4064/aa161-2-1

Keywords: determine equidistribution property possibly moving targets holds rational function degree projective line algebraically closed field characteristic complete respect non trivial absolute value characterization could useful positive characteristic based variational argument purely local proof adelic equidistribution theorem possibly moving targets which due favre rivera letelier using dynamical diophantine approximation theorem silverman szpiro tucker proof general equidistribution theorem possibly moving targets which due lyubich archimedean favre rivera letelier constant targets non archimedean characteristic moving targets non archimedean zero characteristic