Sur les bornes inférieures de l'écart des variétés invariantes

Let m be an integer ⩾3, set ℏ(r)=12(r12+⋯+rm−12)+rm for r∈Rm, and consider a badly approximable vector ω¯0∈Rm−2. Fix α>1, L>0 and R>1+‖ω¯0‖∞. We construct a sequence (HN) of Gevrey-(α,L) Hamiltonian functions of Tm×B¯∞(0,R), which converges to ℏ when N→∞, such that for each N the system generated by HN possesses a (m−1)-dimensional hyperbolic invariant torus with fixed frequency vector (ω¯0,1), which admits a homoclinic point with splitting matrix of the form diag(0,νN,…,νN,0)∈Mm(R), with νN⩾exp(−c(1ɛN)12(α−1)(m−2)), where ɛN:=‖HN−ℏ‖α,L and c>0. To cite this article: J.-P. Marco, C. R. Acad. Sci. Paris, Ser. I 340 (2005).