×

A bumpy metric theorem and the Poisson relation for generic strictly convex domains. (English) Zbl 0713.58005

The paper contains two results. The first one generalizes the bumpy metric theorem of Abraham (which asserts that for every \(C^{\infty}\)- manifold M there is a residual set of smooth metrics which are bumpy in the sense that every closed geodesic is nondegenerate): for generic submanifolds of \({\mathbb{R}}^ n\) without boundary the induced standard metric is a bumpy metric. The second result is a consequence of the first one: for generic bounded strictly convex domains \(\Omega \subset {\mathbb{R}}^ m\), \(m\geq 2\), the singular support of the tempered distribution \(\sigma (t)=\sum^{\infty}_{j=1}\cos \lambda_ jt\) (where \(\{\lambda^ 2_ j\}\) denotes the spectrum of the Dirichlet problem on \(\Omega\}\) coincides with the “length spectrum” (corresponding to all generalized periodic geodesics in \(\Omega\)).
Reviewer: H.Schröder

MSC:

58D17 Manifolds of metrics (especially Riemannian)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
53C22 Geodesics in global differential geometry
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Abraham, R.: Bumpy metrics. Global analysis. Proc. Symp. Pure Math.14, 1-3 (1970) · Zbl 0215.23301
[2] Abraham, R., Marsden, J.: Foundations of mechanics. London, Amsterdam: Benjamin/Cummings 1978 · Zbl 0393.70001
[3] Abraham, R., Robbin, J.: Transversal mappings and flows. New York: Benjamin 1967 · Zbl 0171.44404
[4] Anderson, K., Melrose, R.: The propagation of singularities along gliding rays. Invent. Math.41, 197-232 (1977) · Zbl 0373.35053 · doi:10.1007/BF01403048
[5] Anosov, D.: On generic properties of closed geodesics. Izv. Acad. Nauk SSSR, Ser. Mat.46, 675-703 (1982); Engl. Transl.: Math. USSR Izv.21, 1-29 (1983) · Zbl 0512.58014
[6] Arnold, V., Avez, A.: Problemes ergodiques de la mecanique classique. Pairs: Gautier-Villars 1967
[7] Chazarain, J.: Formule de Poisson pour les varietes riemanniennes. Invent. Math.24, 65-82 (1974) · Zbl 0281.35028 · doi:10.1007/BF01418788
[8] Colin de Verdiere, Y.: Sur les longueurs des trajectorires periodiques d’un billiard, pp. 122-139 dans Geometrie Symplectique et de Contact: Autour du Theoreme de Poincere-Birkhoff. Paris: Hermann 1984
[9] Duistermaat, J., Guillemin, V.: The spectrum of positive elliptic operators and periodic geodesics. Invent. Math.29, 39-79 (1975) · Zbl 0307.35071 · doi:10.1007/BF01405172
[10] Golubitsky, M., Guillemin, V.: Stable mappings and their singularities. Berlin Heidelberg New York: Springer 1973 · Zbl 0294.58004
[11] Guillemin, V., Melrose, R.: The Poisson summation formula for manifolds with boundary. Adv. Math.32, 204-232 (1979) · Zbl 0421.35082 · doi:10.1016/0001-8708(79)90042-2
[12] Guillemin, V., Melrose, R.: A cohomological invariant of discrete dynamical systems, pp. 672-679 in Christoffel Centennial Volume, Putzer, P.I., Feher, F. (eds.) Basel: Birkh?user 1981 · Zbl 0482.58032
[13] Hirsch, M.: Differential topology. Berlin Heidelberg New York: Springer 1976 · Zbl 0356.57001
[14] Klingenberg, W., Lectures on closed geodesics. Berlin Heidelberg New York: Springer 1978 · Zbl 0397.58018
[15] Klingenberg, W., Takens, F.: Generic properties of geodesic flows. Math. Ann.197, 323-334 (1972) · Zbl 0225.58006 · doi:10.1007/BF01428204
[16] Magnuson, A.: Symplectic singularities, periodic orbits of the billiard ball map, and the obstacle problem. Thesis, Massachusetts Institute of Technology, 1984
[17] Marvizi, S., Melrose, R.: Spectral invariants of convex planar regions. J. Differ. Geom.17, 475-502 (1982) · Zbl 0492.53033
[18] Melrose, R., Sj?strand, J.: Singularities in boundary value problems I. Commun. Pure Appl. Math.31, 593-617 (1978) · Zbl 0378.35014 · doi:10.1002/cpa.3160310504
[19] Meyer, K., Palmore, J.: A generic phenomenon in conservative Hamiltonian systems. Global analysis. Proc. Symp. Pure Math.14, 185-189 (1970) · Zbl 0217.20801
[20] Petkov, V., Stojanov, L.: Periods of multiple reflecting geodesics and inverse spectral results. Am. J. Math.109, 619-668 (1987) · Zbl 0652.35027 · doi:10.2307/2374608
[21] Petkov, V., Stojanov, L.: Spectrum of the Poincare map for periodic reflecting rays in generic domains. Math. Z.194, 505-518 (1987) · Zbl 0673.58035 · doi:10.1007/BF01161919
[22] Petkov, V., Stojanov, L.: On the number of periodic reflecting rays in generic domains. Ergodic Theory Dyn. Syst.8, 81-91 (1988) · Zbl 0668.58005 · doi:10.1017/S0143385700004338
[23] Clark Robinson R.: Generic properties of conservative systems II. Am. J. Math.92, 897-906 (1970) · Zbl 0212.56601 · doi:10.2307/2373401
[24] Takens, F.: Hamiltonian systems: Generic properties of closed orbits and local perturbations. Math. Ann.188, 304-312 (1970) · Zbl 0196.27003 · doi:10.1007/BF01431464
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.