Arnold diffusion in perturbations of analytic integrable Hamiltonian systems
Very interesting and detailed paper addressing the following problem: is a given analytic integrable Hamiltonian the limit of analytic Hamiltonians with Arnold diffusion along a path of length uniformily bounded from below? The main result of the paper answers this question in the case of an integrable analytic Hamiltonian in \(\mathbb{T}^n\times \mathbb{R}^n\), \(H_0(\theta,r)=H_0(r)\) such that \(\frac{\partial}{\partial r}H_0\) is a local diffeomorphism. A suitable path from \(H_0\) is chosen in the space of the action variables. Then a curve of analytic Hamiltonians (arbitrary small analytic perturbations of \(H_0\)) is constructed such that any Hamiltonian on it (except the integrable one) has Arnold diffusion. Since having Arnold diffusion is an open property, there is an open neighbourhood of the curve (in the analytic topology) of Hamiltonians having Arnold diffusion. The complete proof is carefully sketched in Section 2 while Sections 3 and 4 are devoted to the proof of auxialiary results, such as the existence and analyticity of invariant manifolds for partially hyperbolic tori and an inclinatiom lemma, nedeed to complete the proof of the main result.
scientific article; zbMATH DE number 1882963