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: $$ H(x,y)=h(y)+eps P(x,y), qquad xinmathbb{T}^n, yinmathbb{R}^n,quad ngeq 3. $$ Under typical perturbation $eps P$, the system admits connecting orbit that passes through any finitely many prescribed small balls in the same energy level $H^{-1}(E)$ provided $E>min h$.