Paternain, Gabriel P. 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]. Reviewer: William J.Satzer jun.(St.Paul) 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 Keywords:hyperbolic dynamics; Euler-Lagrange flows; Anosov energy level; critical value of the Lagrangian PDFBibTeX XMLCite \textit{G. P. Paternain}, in: Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D'Hères: Univ. de Grenoble I, Institut Fourier. 127--151 (1997; Zbl 0898.58041) Full Text: EuDML