Collective geodesic flows

[Flots géodésiques collectifs]

Butler, Léo T. ; Paternain, Gabriel P.

Annales de l'Institut Fourier, Tome 53 (2003), p. 265-308 / Harvested from Numdam

Résumé

On démontre que la plupart des groupes de Lie semi-simples et compacts, admettent plusieurs métriques riemanniennes invariantes à gauche dont le flot géodésique possède une entropie topologique positive. De plus, on démontre que, sur la plupart des espaces homogènes, il existe dans chaque voisinage de la métrique bi-invariante, des métriques riemanniennes "collectives", dont le flot géodésique possède une entropie topologique positive. On discute des autres propriétés du flot géodésique collectif.

Abstract

We show that most compact semi-simple Lie groups carry many left invariant metrics with positive topological entropy. We also show that many homogeneous spaces admit collective Riemannian metrics arbitrarily close to the bi-invariant metric and whose geodesic flow has positive topological entropy. Other properties of collective geodesic flows are also discussed.

Détail

Publié le : 2003-01-01

MR 1973073 | Zbl 1066.53135

DOI : https://doi.org/10.5802/aif.1944
Classification:  53D25,  37D40,  37B40,  53D20
Mots clés: flots géodésiques collectifs, entropie topologique, algèbres de Lie semi-simples, application du moment, intégrale de Melnikov

Bibliographie

[9] V. Guillemin; S. Sternberg The moment map and collective motion, Ann. Physics, Tome 127 (1980), pp. 220-253 | Article | MR 576424 | Zbl 0453.58015

[1] M. Adler; P. Van; Moerbeke Geodesic flow on $\rm SO(4)$ and the intersection of quadrics, Proc. Nat. Acad. Sci. U.S.A, Tome 81 (1984), pp. 4613-4616 | Article | MR 758421 | Zbl 0545.58027

[2] M. Adler; P. Van; Moerbeke The algebraic integrability of geodesic flow on $so(4)$, Invent. Math, Tome 67 (1982), pp. 297-331 | Article | MR 665159 | Zbl 0539.58012

[3] V. I. Arnold Dynamical Systems III, Springer Verlag, Berlin, Encyclopaedia of Mathematical Sciences (1988) | MR 1292465 | Zbl 0623.00023

[4] O. I. Bogoyavlensky Integrable Euler equations on $so(4)$ and their physical applications, Comm. Math. Phys, Tome 93 (1984), pp. 417-436 | Article | MR 745694 | Zbl 0567.58012

[5] A.V. Bolsinov; A.T. Fomenko Orbital isomorphism between two classical integrable systems. The Euler case and the Jacobi problem, Lie groups and Lie algebras, Kluwer Acad. Publ., Dordrecht (Math. Appl) Tome 433 (1998), pp. 359-382 | Zbl 0904.58024

[6] R. Bowen Entropy for Group Endomorphisms and Homogeneous spaces, Trans. of Am. Math. Soc, Tome 153 (1971), pp. 401-414 | Article | MR 274707 | Zbl 0212.29201

[7] K. Burns; H. Weiss A geometric criterion for positive topological entropy, Comm. Math. Phys, Tome 172 (1995), pp. 95-118 | Article | MR 1346373 | Zbl 0945.37003

[8] V. Guillemin; S. Sternberg Symplectic techniques in physics, Cambridge University Press, Cambridge (1984) | MR 770935 | Zbl 0576.58012

[10] L. Haine The algebraic complete integrability of geodesic flow on $\rm so(N)$, Comm. Math. Phys, Tome 94 (1984), pp. 271-287 | Article | MR 761797 | Zbl 0584.58023

[11] H. Hancock Lectures on the Theory of Elliptic Functions, Dover, New York (1909 (reprint 1958))

[12] B. Hasselblatt; A. Katok Introduction to the modern theory of dynamical systems, Cambridge University Press, Encyclopedia of Mathematics and its Applications, Tome 54 (1995) | MR 1326374 | Zbl 0878.58020

[13] S. Helgason Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, Providence, RI (2001. Corrected reprint of the 1978 original) | MR 1834454 | Zbl 0993.53002

[14] P. Holmes; J. Marsden Horseshoes and Arnol$'$d diffusion for Hamiltonian systems on Lie groups, Indiana Univ. Math. J, Tome 32 (1983), pp. 273-309 | Article | MR 690190 | Zbl 0488.70006

[15] W. Klingenberg Riemannian Geometry, De Gruyter, Berlin-New York (1982) | MR 666697 | Zbl 0495.53036

[16] V.V. Kozlov; D.A. Onishchenko Nonintegrability of Kirchhoff's equations (Russian), Dokl. Akad. Nauk SSSR, Tome 266 (1982), pp. 1298-1300 | MR 681629 | Zbl 0541.70009

[17] S. V. Manakov A remark on the integration of the {E}ulerian equations of the dynamics of an n-dimensional rigid body, Funkcional. Anal. i Priložen., Tome 10 (1976), p. 93-94 | MR 455031 | Zbl 0343.70003

[18] R. Mañé On the topological entropy of geodesic flows, J. Diff. Geom, Tome 45 (1997), pp. 74-93 | MR 1443332 | Zbl 0896.58052

[19] A. S. Miščenko Integrals of geodesic flows on Lie groups, Funkcional. Anal. i Priložen., Tome 4 (1970), pp. 73-77 | MR 274891

[20] A. S. Miščenko; A. T. Fomenko The integration of Euler equations on a semisimple Lie algebra, Dokl. Akad. Nauk SSSR, Tome 231 (1976), pp. 536-538 | MR 501139 | Zbl 0392.58001

[21] A. S. Miščenko; A. T. Fomenko A generalized Liouville method for the integration of Hamiltonian systems, Funkcional. Anal. i Priložen., Tome 12 (1978), pp. 46-56 | MR 516342 | Zbl 0396.58003

[22] O.E. Orel Euler-Poinsot dynamical systems and geodesic flows of ellipsoids: topologically nonconjugation, Tensor and vector analysis, Gordon and Breach, Amsterdam (1998), pp. 76-84 | Zbl 0935.37020

[23] C. Robinson Horseshoes for autonomous Hamiltonian systems using the Melnikov integral, Ergodic Theory Dynam. Systems (Charles Conley Memorial Issue) Tome 8* (1988), pp. 395-409 | Zbl 0666.58039

[24] A. Thimm Integrable geodesic flows on homogeneous spaces, Ergod. Th. and Dyn. Syst, Tome 1 (1981), pp. 495-517 | MR 662740 | Zbl 0491.58014

[25] A.P. Veselov Conditions for the integrability of Euler equations on $so(4)$, (Russian), Dokl. Akad. Nauk SSSR, Tome 270 (1983), pp. 1298-1300 | MR 712935 | Zbl 0539.58013

[26] P. Walters An introduction to ergodic theory, Springer-Verlag, New York-Heidelberg-Berlin, Graduate Texts in Mathematics (1982) | MR 648108 | Zbl 0475.28009

[27] A. Weinstein The local structure of Poisson manifolds, J. Differential Geom, Tome 18 (1983), pp. 523-557 | MR 723816 | Zbl 0524.58011

[28] Z. Xia Homoclinic points and intersections of Lagrangian submanifolds, Discrete Contin. Dynam. Systems, Tome 6 (2000), pp. 243-253 | Article | MR 1739927 | Zbl 1009.37040