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Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows. (English) Zbl 0667.58050
We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to the problem of rigidity of geodesic flows for Riemannian manifolds of negative curvature.
Reviewer: R.Feres

MSC:
37D99 Dynamical systems with hyperbolic behavior
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
53C20 Global Riemannian geometry, including pinching
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References:
[1] Mather, Notes on Topological Stability (1970)
[2] Mañé, Ergodic Theory and Differentiable Dynamics (1987) · doi:10.1007/978-3-642-70335-5
[3] Feres, Rigidity of geodesic flows on negatively curved manifolds of dimensions 3 and 4
[4] Kanai, Tensorial ergodicity of geodesic flows · Zbl 0679.58034 · doi:10.1007/BFb0083053
[5] Kanai, Ergod. Th. & Dynam. Sys. 8 pp 215– (1988)
[6] Klingenberg, Riemannian Geometry 1 pp none– (1982)
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