Uniform hyperbolicity of invariant cylinder

For a nearly integrable Hamiltonian systems $H=h(p)+ϵP(p,q)$ with $(p,q)\in {\mathbb{R}}^3\times {\mathbb{T}}^3$, large normally hyperbolic invariant cylinders exist along the whole resonant path, except for the ${\sqrt{ϵ}}^{1+d}$-neighborhood of finitely many double resonant points. It allows one to construct diffusion orbits to cross double resonance.