by Christophe Golé (Smith College, USA)

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