  Let X be a variety over a number field and let f: X --> X be an "interesting"
rational self-map with a fixed point q. We make some general remarks concerning
the possibility of using the behaviour of f near q to produce many rational
points on X. As an application, we give a simplified proof of the potential
density of rational points on the variety of lines of a cubic fourfold
(originally obtained by Claire Voisin and the first author in 2007).
