Moser's theorem for hyperbolic-type degenerate lower tori in Hamiltonian system

In this paper, we give a Moser-type theorem for C-l-smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = <omega, y > + 1/2v(2) - u(2d) + P (x, y, u, v), (x, y, u, v) is an element of T-n x R-n x R-2, which is associated with the standard symplectic structure, with d >= 1. Due to the difficulty coming from the degeneracy, our result is quite different from L. Chierchia and D. Qian's work [8] (non-degenerate case). An interesting phenomenon shown in degenerate case is the l-regularity of above Hamiltonian system not only relies on the tori's dimension n but also strongly relies on the degenerate index d. Under arbitrary small perturbation P, we prove that if l >= (5d + 2)(8 tau + 3), where tau > n - 1, the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C-beta for any beta <= 8 tau + 2. Our result can be seen a generalization of paper [42] from analytic case to C-l-smooth case and can also be seen a generalization of paper [8] from non-degenerate case to degenerate case. (C) 2021 Elsevier Inc. All rights reserved.