The transfer lemma for Graff tori and Arnold diffusion time

Let $H_\mu, 0 < \mu < < 1$ be a small perturbation of size $\mu$ of an initially hyperbolic Hamiltonian system. We prove that Graff tori satisfy the transitivity property : if $T_1, T_2$ and $T_3$ are three Graff tori such that $W^+(T_1)$ (resp. $W^+(T_2))$ and $W^-$ $(T_2)$ (resp. $W^-$ $(T_3))$ intersect transversally in a given energy level $H$ with an angle of order $\mu$, then $W^+(T_1)$ and $W^-$ $(T_3)$ intersect transversally in $H$ with an angle of order $\mu- c(\mu)$, with $c(\mu)$ exponentially small in $\mu$. The proof is based on a quantitative version of the $\lambda$-lemma for Graff tori called the transfer lemma. This result allows us to compute the Arnold diffusion time along transition chains for initially hyperbolic Hamiltonian systems. We prove that this time is polynomial in the inverse of the perturbing parameter.