Sevryuk M.B.
Invariant sets of degenerate Hamiltonian systems near equilibria
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ABSTRACT. For any collection of $n\geq 2$ 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 R\"ussmann and
Anosov--Katok theorems, and a recent theorem by Herman.