  These are notes for a minicourse given at Regional Norte UdelaR in Salto,
Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian
Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative
twist maps of the 2-dimensional annulus and focus on what happens close to the
Aubry-Mather sets: definition of the Green bundles, link between hyperbolicity
and shape of the Aubry-Mather sets, behaviour close to the boundaries of the
instability zones. We will also give some open questions. This course is the
second part of a minicourse that was begun by R. Potrie. Some topics of the
part of R. Potrie will be useful for this part. Many thanks to E. Maderna and
L. Rifford for the invitation to give the mini-course and to R. Potrie for
accepting to share the course with me.
