A paradifferential approach for hyperbolic dynamical systems and applications

. We develop a paradifferential approach for studying non-smooth hyperbolic dynamics on manifolds and related non-linear PDE from a microlocal point of view. As an application, we describe the microlocal regularity, i.e the H s wave-front set for all s , of the unstable bundle E u for an Anosov flow. We also recover rigidity results of Hurder-Katok and Hasselblatt in the Sobolev class rather than H¨older: there is s 0 > 0 such that if E u has H s regularity for s > s 0 then it is smooth (with s 0 = 2 for volume preserving 3-dimensional Anosov flows). It is also shown in the Appendix that it can be applied to deal with non-smooth flows and potentials. This work could serve as a toolbox for other applications.