Homotopical dynamics IV. Hopf invariants and Hamiltonian flows

In a non-compact context the first natural step in the search for periodicorbits of a hamiltonian flow is to detect bounded ones. In this paper we showthat, in a non-compact setting, certain algebraic topological constraintsimposed to a gradient flow of the hamiltonian function $f$ imply the existenceof bounded orbits for the hamiltonian flow of $f$. Once the existence ofbounded orbits is established, under favorable circumstances, application ofthe $C^{1}$-closing lemma leads to periodic ones.