On the Lebesgue measure of the periodic points of a contact manifold. (English) Zbl 0816.58008
This paper is concerned with the Lebesgue measure of the periodic points of contact vector fields on contact manifolds. For any smooth contact manifold we prove that the Lebesgue measure of the set of the periodic points is equal to that of the absolutely periodic points. As a consequence we obtain that for any analytic connected and complete contact manifold either the Lebesgue measure of the periodic points is zero or the flow of the corresponding vector field is totally periodic. A similar result is proved for the periodic points of a Hamiltonian vector field on an energy surface of a contact type. Applications in the semi- classical asymptotics of the Schrödinger operator are obtained.
Reviewer: V.Pelkov (Bordeaux)
MSC:
| 58C35 | Integration on manifolds; measures on manifolds |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
References:
| [1] | Brummelhuis, R., Uribe A.: A semiclassical trace formula for Schrödinger operators. Commun. Math. Phys.136, 567–584 (1991) · Zbl 0729.35093 · doi:10.1007/BF02099074 |
| [2] | Françoise, J.P., Guillemin, V.: On the period spectrum of a symplectic map. J. Funct. Anal.100, 317–358 (1991) · Zbl 0739.58020 · doi:10.1016/0022-1236(91)90114-K |
| [3] | Helffer B., Knauf A., Siedentop H., Weikard R.: On the absence of a first order correction for the number of bound states of a schrödinger operator with coulomb singularity. Commun. Partial Differ. Equations17, 615–639 (1992) · Zbl 0756.35061 |
| [4] | Helffer B., Martinez A., Robert D.: Ergodicité et limite semi-classique. Commun. Math. Phys.109, 313–326 (1987) · Zbl 0624.58039 · doi:10.1007/BF01215225 |
| [5] | Helffer B., Robert D.: Propriétés asymptotiques du spectre d’opérateurs pseudodifférentieles surR n . Commun partial Differ. Equations7, 795–882 (1982) · Zbl 0501.35081 · doi:10.1080/03605308208820239 |
| [6] | Helffer B., Robert D.: Calcul fonctionel par la transformation de Mellin. J. Funct. Anal.53, 246–268 (1983) · Zbl 0524.35103 · doi:10.1016/0022-1236(83)90034-4 |
| [7] | Hörmander, L.: The Analysis of Linear Partial Differential Operators III. Berlin-Heidelberg-New York. Springer 1985 · Zbl 0601.35001 |
| [8] | Ivrii, V.: Semiclassical microlocal analysis and precise spectral asymptotics. Ecole Polytechnique, Preprint 2, 1991 |
| [9] | Petkov, V., Robert, D.: Asymptotiques semi-clasques du spectre d’hamiltoniens quantiques et trajectoires classiques périodiques. Commun. Partial Differ. Equations10, 4, 365–390 (1985) · Zbl 0574.35067 · doi:10.1080/03605308508820382 |
| [10] | Petkov, V., Safarov Yu.: Measure of the periodic points of analytic Hamiltonian flows. Preprint, 1990 |
| [11] | Popov, G.: Length spectrum invariants of Riemannian manifolds. Math. Z.213, 311–351 (1993) · Zbl 0804.53068 · doi:10.1007/BF03025724 |
| [12] | Safarov Yu., Vasil’ev D.: Branching Hamiltonian billiards. Dokl. Akad. Nauk SSSR301, 271–274 (1988): English translation: Sov. Math. Dokl.38, 64–68 (1989) |
| [13] | Viterbo, C.: A proof of Weinstein’s conjecture. Ann. Inst. Henri Poincaré (Analyse nonlinéaire)4, 337–356 (1987) · Zbl 0631.58013 |
| [14] | Weinstein, A.: On the hypotheses of Rabinowitz’ periodic orbit theorems. J. Differ. Equations 33, 353–358 (1979) · doi:10.1016/0022-0396(79)90070-6 |
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