Diophantine interval exchange maps

I will review results concerning the dynamics of (generalized) interval exchange maps obtained in collaboration with Moussa and Yoccoz and with Forni and Matheus. I will explain what part of the theory of circle diffeomorphisms extends to interval exchange maps: how to generalize the notion of rotation number and how the diophantine conditions needed to solve the cohomological equation and to prove a linearization theorem relate to the (unavoidable) loss of differentiability
which characterizes these "small divisor" problems.