Arnold diffusion in nearly integrable Hamiltonian systems of arbitrary degrees of freedom

In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: $$H(x,y)=h(y)+\text{eps}P(x,y),x\in {\mathbb{T}}^n,y\in {\mathbb{R}}^n,n\ge 3.$$ Under typical perturbation $\text{eps}P$, the system admits “connecting” orbit that passes through any finitely many prescribed small balls in the same energy level $H^{-1}(E)$ provided $E>\min h$.