  We consider, on a compact manifold, the group of diffeomorphisms that are
isotopic to the identity. We show that every recurrent element is a distorsion
element. This generalizes Avila's theorem on circle diffeomorphisms. The method
also provides a new proof of a result by Calegari and Freedman: on a sphere, in
the group of homeomorphisms that are isotopic to the identity, every element is
distorted.
