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Metrics with positive topological entropy on \(S^ 2\). (Métriques à entropie topologique positive sur \(S^ 2\).) (French) Zbl 0778.58035
Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 10, Année 1991-1992, 97-107 (1992).
It is believed that geodesic flows on compact manifolds with negative curvatures have positive entropy and that they are ergodic. However, Donnay (1989) and Burns and Gerber (1989) proved that to build ergodic metrics with positive entropy on \(S^ 2\) is topologically not possible and that the properties can be obtained for negative curvature cases like their examples. In this paper it is proved that it is not the problem of the curvature and it is possible to construct metrics with positive curvature and positive entropy on \(S^ 2\). However, the problem is still open for Liouville entropy and ergodicity.
Reviewer: Y.Kozai (Tokyo)
MSC:
37A99 Ergodic theory
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
54C70 Entropy in general topology
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