Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders

In the present paper we prove a form of Arnold diffusion. The main result says that for a “generic” perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ $$H_0(p)+\text{eps}{H}_1(\mathrm{th},p,t),\mathrm{th}\in {\text{T}}^n,p\in B^n,t\in \text{T}=\mathbb{R}\mathrm{/}\text{T},$$ with strictly convex $H_0$ there exists an orbit $({\mathrm{th}}_{\text{e}},p_e)(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}|p(t)-p(0) | >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\text{e}$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders of limited regularity, but of large size, extrapolating on \cite{Be3} and \cite{KZZ}. Once these cylinders are constructed we use versions of Mather variational method developed in Bernard \cite{Be1}, Cheng-Yan \cite{CY1, CY2}.