Invariant sets of degenerate Hamiltonian systems near equilibria
It is proved that, for any collection of \(n\) frequencies \(\omega_i\), there exists a \(C^\infty\) Hamiltonian system with \(n\) degrees of freedom which has an elliptic equilibrium at 0, \(\omega_i\) are the corresponding eigenfrequencies, the Hamiltonian vector field is linear up to a remainder which is flat at 0, and the measure of the set of the \(n\)-dimensional invariant tori in a \(\varepsilon\)--neighborhood of 0 tends to zero with \(\varepsilon\), faster than any prescribed function. The same result is valid for symplectic maps and also for reversible flows and maps.
scientific article; zbMATH DE number 1340946