October  2001, 7(4): 787-800. doi: 10.3934/dcds.2001.7.787

The transfer lemma for Graff tori and Arnold diffusion time

1. 

Equipe de Mathématiques de Besançon, CNRS-UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon cedex, France

Received  July 1999 Revised  May 2001 Published  July 2001

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.
Citation: Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787
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