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On the creation of conjugate points for Hamiltonian systems. (English) Zbl 0999.37037

Summary: “For a fixed Hamiltonian \(H\) on the cotangent bundle of a compact manifold \(M\) and a fixed energy level \(e\), we prove that the set \(A_e\), of potentials \(\varphi\) on \(M\) such that the Hamiltonian flow of \(H+ \varphi\) is Anosov, is the interior in the \(C^2\) topology of the set \({\mathcal B}_e\) of potentials such that the flow has no conjugate points”. This theory extends a result of R. O. Ruggiero [Math. Z. 208, 41-55 (1991; Zbl 0749.58042)]. The proof uses results from G. P. and M. Paternain [Math. Z. 217, 367-376 (1994; Zbl 0820.58022)].

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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