G\'eom\'etrie, points rationnels et it\'er\'es des automorphismes de l'espace affine

Abstract

We study over a number field, the iterates of automorphisms of the affine space. More precisely, we are interested in the periodic and non-periodic points; for the former the questions are similar to the ones about torsion points on abelian varieties, for the latter the questions are similar to the problems on counting rational points on varieties. In order to study this problem, we define geometric invariants, which are defined thanks to the cone of divisors associated to some varieties constructed from the automorphisms. We study, in all dimension, these invariants and bound them above for one of them we obtain a bound depending on the dynamical degree, this bound is reach by some automorphisms, in particular this bound is optimal. For an other invariant, we obtain in all dimension, thanks to a geometrical construction and under some geometric conditions, a better bound.