Abstract: In the 80's two type of related rigidity results emerged, one of geometric flavor and the other dynamical, but both have a dynamical background.  Otal and Croke showed that for negatively curved surface the marked length spectrum determines the isometry type of the surface. On the other hand, Shub ans Sullivan for expanding maps and later de la LLave, Marco and Moriyon for Anosov diffeomorphisms in dimension 2 showed that the marked Lyapunov spectrum determines the smooth isomorphism type of the system. In this talk I will discuss new developments along this line of problems, discussing a more general framework where these theories can be developed. This project is joint with A. Gogolev and M. Leguil.