Raphael Douady's thesis, Application du theoreme des tores invariants, has been cited in numerous papers by many experts.

According Wikipedia, he proves of the equivalence of KAM theory for Hamiltonian systems and for symplectomorphisms, opening the gate to discrete KAM theory.

According to the descriptions in various references, he lowers the regularities for Moser twist mapping theorem and Lazutkin's existence of caustics of strictly convex billiards. But there are different versions of his results in different references.

I want to know the exact statements in his thesis, and the methods/outline he used to prove them.

Is there some textbooks/lecturenotes with a self-contained proof of these results now?

Thanks!