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.
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.
Keywords: collective geodesic flows, topological entropy, semi-simple Lie algebras, moment map, Melnikov integral
Mot clés : flots géodésiques collectifs, entropie topologique, algèbres de Lie semi-simples, application du moment, intégrale de Melnikov
Butler, Léo T. 1; Paternain, Gabriel P. 2
@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{\textquoteright}institut Fourier}, volume = {53}, number = {1}, year = {2003}, doi = {10.5802/aif.1944}, zbl = {1066.53135}, mrnumber = {1973073}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1944/} }
TY - JOUR AU - Butler, Léo T. AU - Paternain, Gabriel P. TI - Collective geodesic flows JO - Annales de l'Institut Fourier PY - 2003 SP - 265 EP - 308 VL - 53 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1944/ DO - 10.5802/aif.1944 LA - en ID - AIF_2003__53_1_265_0 ER -
%0 Journal Article %A Butler, Léo T. %A Paternain, Gabriel P. %T Collective geodesic flows %J Annales de l'Institut Fourier %D 2003 %P 265-308 %V 53 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1944/ %R 10.5802/aif.1944 %G en %F AIF_2003__53_1_265_0
Butler, Léo T.; Paternain, Gabriel P. Collective geodesic flows. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 265-308. doi : 10.5802/aif.1944. https://aif.centre-mersenne.org/articles/10.5802/aif.1944/
[9] The moment map and collective motion, Ann. Physics, Volume 127 (1980), pp. 220-253 | DOI | MR | Zbl
[1] Geodesic flow on and the intersection of quadrics, Proc. Nat. Acad. Sci. U.S.A, Volume 81 (1984), pp. 4613-4616 | DOI | MR | Zbl
[2] The algebraic integrability of geodesic flow on , Invent. Math, Volume 67 (1982), pp. 297-331 | DOI | MR | Zbl
[3] Dynamical Systems III, Encyclopaedia of Mathematical Sciences, Springer Verlag, Berlin, 1988 | MR | Zbl
[4] Integrable Euler equations on and their physical applications, Comm. Math. Phys, Volume 93 (1984), pp. 417-436 | DOI | MR | Zbl
[5] Orbital isomorphism between two classical integrable systems. The Euler case and the Jacobi problem, Lie groups and Lie algebras (Math. Appl), Volume 433 (1998), pp. 359-382 | Zbl
[6] Entropy for Group Endomorphisms and Homogeneous spaces, Trans. of Am. Math. Soc, Volume 153 (1971), pp. 401-414 | DOI | MR | Zbl
[7] A geometric criterion for positive topological entropy, Comm. Math. Phys, Volume 172 (1995), pp. 95-118 | DOI | MR | Zbl
[8] Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984 | MR | Zbl
[10] The algebraic complete integrability of geodesic flow on , Comm. Math. Phys, Volume 94 (1984), pp. 271-287 | DOI | MR | Zbl
[11] Lectures on the Theory of Elliptic Functions, Dover, New York, 1909 (reprint 1958)
[12] Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995 | MR | Zbl
[13] Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original | MR | Zbl
[14] Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana Univ. Math. J, Volume 32 (1983), pp. 273-309 | DOI | MR | Zbl
[15] Riemannian Geometry, De Gruyter, Berlin-New York, 1982 | MR | Zbl
[16] Nonintegrability of Kirchhoff's equations (Russian), Dokl. Akad. Nauk SSSR, Volume 266 (1982), pp. 1298-1300 | MR | Zbl
[17] A remark on the integration of the {E}ulerian equations of the dynamics of an n-dimensional rigid body, Funkcional. Anal. i Priložen., Volume 10 (1976), pp. 93-94 | MR | Zbl
[18] On the topological entropy of geodesic flows, J. Diff. Geom, Volume 45 (1997), pp. 74-93 | MR | Zbl
[19] Integrals of geodesic flows on Lie groups, Funkcional. Anal. i Priložen., Volume 4 (1970), pp. 73-77 | MR
[20] The integration of Euler equations on a semisimple Lie algebra, Dokl. Akad. Nauk SSSR, Volume 231 (1976), pp. 536-538 | MR | Zbl
[21] A generalized Liouville method for the integration of Hamiltonian systems, Funkcional. Anal. i Priložen., Volume 12 (1978), pp. 46-56 | MR | Zbl
[22] Euler-Poinsot dynamical systems and geodesic flows of ellipsoids: topologically nonconjugation, Tensor and vector analysis (1998), pp. 76-84 | Zbl
[23] Horseshoes for autonomous Hamiltonian systems using the Melnikov integral, Ergodic Theory Dynam. Systems (Charles Conley Memorial Issue), Volume 8* (1988), pp. 395-409 | Zbl
[24] Integrable geodesic flows on homogeneous spaces, Ergod. Th. and Dyn. Syst, Volume 1 (1981), pp. 495-517 | MR | Zbl
[25] Conditions for the integrability of Euler equations on , (Russian), Dokl. Akad. Nauk SSSR, Volume 270 (1983), pp. 1298-1300 | MR | Zbl
[26] An introduction to ergodic theory, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg-Berlin, 1982 | MR | Zbl
[27] The local structure of Poisson manifolds, J. Differential Geom, Volume 18 (1983), pp. 523-557 | MR | Zbl
[28] Homoclinic points and intersections of Lagrangian submanifolds, Discrete Contin. Dynam. Systems, Volume 6 (2000), pp. 243-253 | DOI | MR | Zbl
Cited by Sources: