Periodic orbits of Hamiltonian systems linear and hyperbolic at infinity
Let the Euclidean space \(\mathbb R^{2n}\) be endowed with the standard symplectic form \(\omega\). The paper is concerned with the dynamics of one-time Hamiltonians that are quadratic at infinity, in particular periodic orbits.NEWLINENEWLINELet \(Q\) be a nondegenerate quadratic form on \(\mathbb R^{2n}\). Then, the Hamiltonian vector field \(X_Q\) is a linear endomorphism of \(\mathbb R^{2n}\), whose eigenvalues are called eigenvalues of \(Q\). The form \(Q\) is hyperbolic if it has no purely imaginary eigenvalues.NEWLINENEWLINEConsider a one time periodic Hamiltonian, i.e., a smooth function \(H:S^1 \times \mathbb R^{2n}\to \mathbb R\). It is quadratic at infinity if \(H\) coincides with \(Q\) outside some open ball in \(\mathbb R^{2n}\). The author proves the following;NEWLINENEWLINETheorem. Let \(H:S^1 \times \mathbb R^{2n}\to \mathbb R\) be a Hamiltonian which is quadratic at infinity and such that \(Q\) has only real eigenvalues. Assume that the Hamiltonian diffeomorphism \(\varphi_H\) has a nondegenerate fixed point \(x\) with nonzero mean index and and \(Fix (\varphi_H) \) is finite. Then, \(\varphi_H\) has simple (that is uniterated) periodic orbits of arbitrarily large period.NEWLINENEWLINERecall that the mean index of a periodic orbit can be found, e.g., in the book of \textit{Y. Long} [Index theory for symplectic paths with applications. Basel: Birkhäuser (2002; Zbl 1012.37012)]. There are many technical ingredients involved here like Floer homology, maximum principle, slow homotopy, which can be found in the paper under review and references therein. In fact, the nondegeneracy condition for \(x\) is weakened by a more general one that \(x\) is isolated and homologically nontrivial (in the sense of Floer homology (see Theorem 4.1)). A better result in low dimensions is obtained. Relations with Conley and Hofer-Zehnder conjectures connecting existence of sufficiently many fixed points to the existence of infinitely many periodic orbits on symplectic manifolds, see [\textit{H. Hofer} and \textit{E. Zehnder}, Symplectic invariants and Hamiltonian dynamics. Basel: Birkhäuser (1994; Zbl 0805.58003)], and also some generalizations of the results, are discussed.
scientific article; zbMATH DE number 6369632