21-16 Qinbo Chen, Rafael de la Llave

Analytic genericity of diffusing orbits in mph{a priori} unstable Hamiltonian systems (456K, PDF) Mar 12, 21

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Abstract. We study the problem of instability in the following mph{a priori} unstable Hamiltonian system with a time-periodic perturbation \[\mathcal{H}_arepsilon(p,q,I,arphi,t)=h(I)+\sum_{i=1}^n\pm \left(rac{1}{2}p_i^2+V_i(q_i) ight)+arepsilon H_1(p,q,I,arphi, t), \] where $(p,q)\in \mathbb{R}^n imes\mathbb{T}^n$, $(I,arphi)\in\mathbb{R}^d imes\mathbb{T}^d$ with $n, d\geq 1$, $V_i$ are Morse potentials, and $arepsilon$ is a small non-zero parameter. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations $H_1$. Indeed, the set of admissible $H_1$ is $C^\omega$ dense and $C^3$ open. The proof also works for arbitrarily small $V_i$. Our perturbative technique for the genericity is valid in the $C^k$ topology for all $k\in [3,\infty)

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