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On Anosov energy levels of Hamiltonians on twisted cotangent bundles. (English) Zbl 0829.58018

A twisted symplectic structure on \(T^* M\) consists of the canonical form to which a magnetic term is added, i.e., the pullback of a closed 2- form on \(M\). A convex Hamiltonian means that \(H(q,\;)\) is convex on \(T^*_q M\) for each \(q\) in \(M\). Given a compact regular energy level, a continuous invariant subbundle is a subbundle \(E\) which is invariant under the differential of the flow and is such that every \(E(\theta)\) is a Lagrangian subspace. Here \(E\) is a subbundle of \(T(T^* M)\) restricted to the energy level.
From the abstract: “When \(\dim M\geq 3\), it is known that such energy level projects onto the whole manifold \(M\), and that \(E\) is transversal to the vertical subbundle. Here we study the case \(\dim M=2\), proving that the projection property still holds, while the transversality property may fail. However, we prove that in the case when \(E\) is the stable or unstable subbundle of an Anosov flow, both properties hold”.

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37D99 Dynamical systems with hyperbolic behavior
70H05 Hamilton’s equations
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