Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems (Q556349)

The author considers a real analytic family of Hamiltonian systems of the form \[ H(x,y,u)= N(y,u)+ P(x,y,u),\tag{1} \] where \((x,y,u)\) lies in a complex neighborhood \(\{(x,y,u)\mid|\text{Im\,}x|\leq \tau\), \(\text{dist}(y, G)\leq s\), \(|u|\leq s\}\) of \(\mathbb T^n\times G\times\{0\}\subset \mathbb T^n\times\mathbb R^n\times \mathbb R^{2m}\), \(G\subset\mathbb R^n\), \(n\geq 2\), is a bounded closed region and \(P\) is small. Besides these, the author assumes, that \[ N_u(y,0)= 0,\quad\det\,N_{uu}(y, 0)\neq 0. \] The main goal is to study the persistence and tangent frequencies preservation of lower-dimensional invariant tori on smooth submanifolds for (1). The author shows that the surviving tori might be elliptic, hyperbolic or of mixed type.