Arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders

Abstract

We prove a form of Arnold diffusion in the a-priori stable case. Let $$H_0(p)+\mathit{ϵ}{H}_1(\mathit{\theta },p,t),\mathit{\theta }\in {\mathbb{T}}^n,p\in B^n,t\in \mathbb{T}=\mathbb{R}/\mathbb{T},$$be a nearly integrable system of arbitrary degrees of freedom $n⩾2$ with a strictly convex H₀. We show that for a “generic” $\mathit{ϵ}{H}_1$, there exists an orbit $(\mathit{\theta },p)$ satisfying $$\Vert p(t)-p(0)\Vert >l(H_1)>0,$$where $l(H_1)$ is independent of $\mathit{ϵ}$. The diffusion orbit travels along a codimension-1 resonance, and the only obstruction to our construction is a finite set of additional resonances.

For the proof we use a combination of geometric and variational methods, and manage to adapt tools which have recently been developed in the a-priori unstable case.