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

Abstract

In this paper, we give a Moser-type theorem for C ℓ -smooth hyperbolic-type degenerate Hamiltonian system with the following Hamiltonian H = 〈 ω , y 〉 + 1 2 v 2 − u 2 d + P ( x , y , u , v ) , ( x , y , u , v ) ∈ T n × R n × 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 ≥ ( 5 d + 2 ) ( 8 τ + 3 ) , where τ > n − 1 , the above hyperbolic-type degenerate Hamiltonian system admits lower dimensional Diophantine tori which are proved to be of class C β for any β ≤ 8 τ + 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.