Generalization of geometric Poincaré theorem for small perturbations
The number of periodic orbits is estimated which persist under a \(C^2\)-smooth perturbation of a Hamiltonian system containing in a level of the Hamiltonian a smooth closed submanifold \(\Lambda\) filled with periodic orbits. It is supposed that the fibration \(\Lambda\) into closed orbits is a smooth bundle with a smooth base \(B\) and \(S^1\) as the fiber. The estimate is given for two cases: i) the bundle is trivial; ii) \(B\) can carry a locally flat Riemannian metric. The estimate of the number of periodic orbits is given via the number of critical points of a smooth function on \(B\). Results obtained are refinement of those proved before by \textit{A. Weinstein} [Math. Z. 159, 235-248 (1978; Zbl 0376.58008), Ann. Math. (2) 98, 377-410 (1973; Zbl 0271.58008)].
scientific article; zbMATH DE number 1281657