Raphael Douady's [thesis](http://citeseer.uark.edu:8080/citeseerx/showciting;jsessionid=82A0592660D37BA6C373CABF62CF034A?cid=2510347), *Applications du théorème des tores invariants*, has been cited in numerous papers by many experts. 


According [Wikipedia](http://en.wikipedia.org/wiki/Raphael_Douady), *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!



Raphael Douady's thesis: Applications du théorème des tores invariants
I'm sorry that my PhD thesis wasn't published indeed and is hard to find (up to my knowledge it is only available at university of Paris 7). As I have a scanned copy of it, I asked people at my software company Riskdata to include it in the list of posted papers, althogh it is not really related to math finance, which I now study. It should be available in the coming days on <https://www.riskdata.com/resources/other>
To answer your question, the optimal degree of differentiability for KAM theorem in dim 2 was obtained by M. Herman and by H Russmann and is equal to 3 for numbers of constant type (|r - p/q| > a/q^2) and 2+epsilon loss of differentiability for other diophantine numbers (to be verified, this is an old result).
Caustics of convex billiards and of external billiards follow the same rule. The regularity of the billiard map is 1 degree less than that of the curve, hence it has to be of class C^4. Again, I'l calling my memory and what I say here must be rechecked!
The relation between KAM for symplectomorphisms and Hamiltonian vector fields is done through Poincaré section and suspension: there is no loss of differentiability and it even works in the analytic setting. So we get the same degree of differentiability for the diffeos and for the vector field, and one more of course for the Hamiltonian itself.
I have another very old unpublished paper (it wasmy master thesis!) where I lower the degree of differentiability required by Moser's proof of KAM in the C^r case, but it is so much suboptimal that I didn't dare to publish it! (I think I lower 500+ to 300+...). Jean-Christopha Yoccoz or Hakan Eliason may know more about the optimal differentiability in higher dimensions.
Good luck!


Raphael Douady


