  For a nearly integrable Hamiltonian systems $H=h(p)+\epsilon P(p,q)$ 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
$\sqrt{\epsilon}^{1+d}$-neighborhood of finitely many double resonant points.
It allows one to construct diffusion orbits to cross double resonance.
