A minimax selector for a class of Hamiltonians on cotangent bundles (Q2706845)

In this paper, the authors generalize a result of \textit{K. Siburg} [Duke Math. J. 92, No. 2, 295-319 (1998; Zbl 1054.37506)] from the case of the cotangent bundle of the torus \(T^n\) to that of the cotangent bundle of a compact orientable Riemannian manifold \(M\). More specifically, they consider \(1\)-periodic Hamiltonians \(H\) on such a cotangent bundle which are quadratic (on each fibre) outside the disk bundle and the associated time-\(1\) diffeomorphisms \(\phi\) of the flow. The main result is then that Hofer's energy \(E(\phi)\) is bounded below by \(\beta(0)\), where \(\beta : H_1(M;\mathbb R) \to \mathbb R\) is defined as in \textit{J. Mather} [Math. Z. 207, 169-207 (1991; Zbl 0696.58027)] by ``minimizing'' a certain action over all probability measures giving a fixed homology class. In this way, the authors connect the dynamical and geometrical viewpoints of Hamiltonian systems.