The Whitney differentiable family of coisotropic invariant tori for a Hamilton system close to degenerate (Q2755247)

Let us consider on the symplectic manifold \((M^{2n},\omega^{2n})\) the Hamilton system with the Hamiltonian \(H(x,\mu)\) depending on a small parameter \(\mu\). Let us suppose that a nonperturbed system with the Hamiltonian \(H(x,0)\) has a family of \(r\)-dimensional coisotropic invariant tori depending on parameters \(u\in U\), \(U\) is a bounded subset of \(\mathbb{R}^{q}\). On the torus \(T_0^{r}\), the motion of the nonperturbed system is quasi-periodic and described by the system of equations \(d\varphi/dt=\omega(u)\), \(\omega:U\to \mathbb{R}^{r}\), where \(\varphi\) is an angular coordinate. The authors deal with the degenerate case \(\omega(u)=(\hat\omega(u), 0)\), \(\hat\omega:U\to\mathbb{R}^{r_0}, r_0<r\). For any sufficiently small \(\mu\), we must find the set \(U_{\mu}\subset U\) such that for any \(u\in U_{\mu}\) there exists a perturbed system of invariant tori \(T_{\mu}^{r}(u)\) close, in some sense, to the torus \(T_{0}^{r}(u)\). The authors prove that for sufficiently small \(\mu\), the set \(U_{\mu}\) has the structure of the Cantor set.