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

Patrick Bernard, Vadim Kaloshin, Ke Zhang

Published 2011-12-13, updated 2011-12-17Version 2

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 $n\ge 2$ \[ H_0(p)+\eps H_1(\th,p,t),\quad \th\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(\th_{\e},p_{e})(t)$ 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 $\e$. 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}.