Symplectic invariants of elliptic fixed points (Q1596291)

This paper is devoted to the dynamics of an area-preserving diffeomorphism in a neighborhood of an elliptic fixed point. The author constructs a new ``local'' symplectic invariant which includes the Birkhoff invariants, but in addition, reflects some part of the dynamical behaviour near the fixed point. To this end the author introduces a variational principle analogous to that in Aubry-Mather theory for monotone twist maps; more precisely he numerizes the Lagrangian action over all orbits with a given rotation number. This yields the minimal action, a function canonically associated to the given symplectic map and invariant under symplectic coordinate changes. The author proves that this function is real valued, and hence a nontrivial invariant. He applies his result to the Reeb flow.