Appears in collection : Summer School 2017 - Arakelov Geometry and diophantine applications

Some themes inspired from number theory have been playing an important role in holomorphic and algebraic dynamics (iteration of rational mappings) in the past ten years. In these lectures I would like to present a few recent results in this direction. This should include: the dynamical Manin-Mumford problem, in particular in the case of product rational maps (P(x),Q(y)) (after Ghioca, Nguyen, and Ye) the “unlikely intersection” problem (after Baker and DeMarco, and also Favre and Gauthier). A key technical tool in these results is the equidistribution theory of points of small height. If time permits, we’ll also discuss the related problem of the equidistribution of roots of random polynomials.