This book concentrates mainly on the theorem of existence of periodic orbits for higher dimensional analogs of Twist maps. The setting is that of a discrete variational calculus and the techniques involve Conley-Zehnder-Morse Theory. They give rise to the concept of ghost tori which are of interest in the dimension 2 case (ghost circles). The debate is oriented somewhat toward the open problem of finding orbits of all (in particular, irrational) rotation vectors.
Twist maps of the annulus; the Aubry-Mather theorem; ghost circles; symplectic twist maps; periodic orbits for symplectic twist maps of Tn x IRn; invariant manifolds; Hamiltonian systems vs. twist maps; periodic orbits for Hamiltonian systems; generalizations of the Aubry-Mather theorem; generating phases and symplectic topology.