Symplectic invariants of elliptic fixed points

Abstract

To the germ of an area-preserving diffeomorphism at an elliptic fixed point, we associate the germ of Mather's minimal action. This yields a strictly convex function which is symplectically invariant and comprises the classical Birkhoff invariants as the Taylor coefficients of its convex conjugate. In addition, however, the minimal action contains information about the local dynamics near the fixed point; for instance, it detects the $C^0$-integrability of the diffeomorphism. Applied to the Reeb flow, this leads to new period spectrum invariants for three-dimensional contact manifolds; a particular case is the geodesic flow on a two-dimensional Riemannian manifold, where the period spectrum is the classical length spectrum.