Collective geodesic flows

Butler, Léo T.; Paternain, Gabriel P.

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.

Classification : 53D25, 37D40, 37B40, 53D20
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

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