For a rational self-map on a projective variety over the field of
algebraic numbers, one can try to study its dynamics (i.e. behavior
under iterates) from a complex geometric point of view, or from an
arithmetic point of view. Each of this two viewpoints contains
interesting results and difficult conjectures/problems. Moreover, the
two viewpoints are closely related. I will briefly introduce the two
viewpoints and their relations. I will also talk about my results (on
both sides) for monomial maps, a family of maps that is more or less
well understood in higher dimension.