Commentarii Mathematici Helvetici

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Volume 75, Issue 4, 2000, pp. 681–700
DOI: 10.1007/s000140050145

Published online: 2000-12-31

Symplectic invariants of elliptic fixed points

K. F. Siburg[1]

(1) Ruhr-Universit├Ąt Bochum, Germany

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 C0--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.

Keywords: Elliptic fixed point, Birkhoff normal form, Aubry--Mather theory, minimal action, Reeb flow, period spectrum, geodesic flow, length spectrum

Siburg K.: Symplectic invariants of elliptic fixed points. Comment. Math. Helv. 75 (2000), 681-700. doi: 10.1007/s000140050145