Invariant sets of degenerate Hamiltonian systems near equilibria

For any collection of $n \geqslant 2$ numbers $\omega_1, \ldots, \omega_n$, we prove the existence of an infinitely differentiable Hamiltonian system of differential equations $X$ with $n$ degrees of freedom that possesses the following properties: 1) $0$ is an elliptic (provided that all the $\omega_i$ are different from zero) equilibrium of system $X$ with eigenfrequencies $\omega_1, \ldots, \omega_n$; 2) system $X$ is linear up to a remainder flat at $0$; 3) the measure of the union of the invariant $n$-tori of system $X$ that lie in the $\varepsilon$-neighborhood of $0$ tends to zero as $\varepsilon \to 0$ faster than any prescribed function. Analogous statements hold for symplectic diffeomorphisms, reversible flows, and reversible diffeomorphisms. The results obtained are discussed in the context of the standard theorems in the KAM theory, the well-known Russmann and Anosov–Katok theorems, and a recent theorem by Herman.