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Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels. (English) Zbl 0898.58041

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 127-151 (1997).
This paper describes recent results obtained via variational methods which concern the dynamics of Euler-Lagrange flows on prescribed energy levels. The author considers Euler-Lagrange flows which are generated by convex and superlinear Lagrangians on closed, connected manifolds. The main result is that if an Anosov energy level has a splitting of class \(C^1\) (i.e., the strong stable and strong unstable bundles are both \(C^1\)), then it must contain minimizing measures with non-zero homology. Furthermore, the energy levels at which the Euler-Lagrange flow is Anosov must be strictly greater than a critical value of the Lagrangian associated with the universal covering, and the energy levels must be free of conjugate points.
For the entire collection see [Zbl 0882.00016].

MSC:

37D99 Dynamical systems with hyperbolic behavior
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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
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