The course will be divided in two parts:
The first part will deal with KAM theory and the aim will be to give a basis of KAM theory in different settings like circle diffeomorphisms, Hamiltonian systems, Schrodinger operators and twist maps of the cylinder. The end of the first part will deal with Aubry-Mather theory for twist maps.

The second part of the course will deal with the Nash-Moser inverse function Theorem on Frechet spaces and applications in geometry and small divisors problems.

Referências:
[1] M.-C Arnaud. Hyperbolicity for conservative twist maps of the 2-dimensional annulus. Publicaciones Matem ́aticas del Uruguay, 2016, Proceedings of the CIMPA research school on Hamiltonian and Lagrangian dynamics, 16, pp.1-39.
[2] V. I. Arnol’d. Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, second edition, 1989.
[3] R. Hamilton The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222.
[4] J. Moser, A rapˆıdly convergent it ́eration method, part II, Ann. Scuola Norm. Sup. di Pisa, Ser. III, 20 (1966), P. 499-535.
[5] S. Marmi. An introduction to small divisors. Quaderni del Dottorato di Ricerca in Matematica, Pisa (2000)
[6] C. E. Wayne. An introduction to KAM theory. In Dynamical Systems and Probabilistic Methods in Partial Differential Equations (Berkeley, CA, 1994), pages 3–29. Amer. Math. Soc., Providence, RI, 1996.
[7] Katok, A. and Hasselblatt, B.: Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54, Cambridge Univer sity Press, Cambridge, 1995.