The goal of this minicourse is to introduce the ideas and techniques involved in the proof of a recent result of Eynard and Navas: the space of Z^d actions on the circle by C^{1+ac} diffeomorphisms is path connected. The precise topics to be treated are:

1) Total variation, Denjoy-Kocsma and the asymptotic variation.
2) Vanishing of the asymptotic variation, the case of the circle and the interval
3) The fundamental inequality: Mather’s invariant vs asymptotic variation. Perturbation lemmas.
4) A proof of path connectedness and open problems.