Collective geodesic flows  [ Flots géodésiques collectifs ]
Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 265-308.

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.

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.

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
@article{AIF_2003__53_1_265_0,
     author = {Butler, L\'eo T. and Paternain, Gabriel P.},
     title = {Collective geodesic flows},
     journal = {Annales de l'Institut Fourier},
     pages = {265--308},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {1},
     year = {2003},
     doi = {10.5802/aif.1944},
     mrnumber = {1973073},
     zbl = {1066.53135},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2003__53_1_265_0/}
}
Butler, Léo T.; Paternain, Gabriel P. Collective geodesic flows. Annales de l'Institut Fourier, Tome 53 (2003) no. 1, pp. 265-308. doi : 10.5802/aif.1944. https://aif.centre-mersenne.org/item/AIF_2003__53_1_265_0/

[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, Encyclopaedia of Mathematical Sciences, Springer Verlag, Berlin, 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 (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, Encyclopedia of Mathematics and its Applications, Tome 54, Cambridge University Press, 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 (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, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 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