Arnold diffusion for nearly integrable Hamiltonian systems

In this paper, we prove that the nearly integrable system of the form H(x,y)=h(y)+εP(x,y),x∈Tn,y∈ℝn,n⩾3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(x,y) = h(y) + \varepsilon P(x,y),\,\,\,\,\,\,\,x \in {\mathbb{T}^n},\,\,\,\,\,\,\,\,y \in {\mathbb{R}^n},\,\,\,\,\,\,n\,\geqslant\,3$$\end{document} admits orbits that pass through any finitely many prescribed small balls on the same energy level H−1(E) provided that E > min h, if h is convex, and εP is typical. This settles the Arnold diffusion conjecture for convex systems in the smooth category. We also prove the counterpart of Arnold diffusion for the Riemannian metric perturbation of the flat torus.