(Arc)-connectedness of the space of Z^d actions on 1-manifolds

We will elaborate on two recent results obtained in collaboration with Hélène Eynard-Bontemps (Inst. Fourier, Grenoble):
- The space of Z^d actions by C^{1+ac} diffeomorphisms of a compact 1-manifold is path-connected;
- The space of Z^d actions by C^2 diffeomorphisms of the interval is connected.
Here, "ac" stands for absolutely continuous. I will deeply comment on several technical problems when dealing with these properties: Mather's homomorphisms, failure of Sternberg's linearization theorem, etc. Several questions will be addressed along the talk.