Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems

We prove the existence of Arnold diffusion in a typical a priori unstable Hamiltonian system outside a small neighbourhood of strong resonances. More precisely, we consider a near-integrable Hamiltonian system with Hamiltonian H = H₀ + εH₁ + O(ε²), where the unperturbed Hamiltonian H₀ is essentially the product of a one-dimensional pendulum and n-dimensional rotator. Coordinates y = (y₁, ..., y_n) on the rotator space are first integrals in the unperturbed system and become slow variables after perturbation.

The main result is as follows. Suppose that the time-periodic perturbation H₁ is C^r-generic, ${\bit r}\in\mathbb{N}\cup \{\infty,\omega\}$ is sufficiently large. A resonance $\langle k,\overline\nu \rangle = 0$, where $\overline\nu = \overline\nu(y)\in\mathbb{R}^{n+1}$ is a frequency vector and $k\in\mathbb{Z}^{n+1}$, is called strong if |k| < C_◊. The constant C_◊ is determined by H₀ and H₁ and does not depend on ε. Let Q be a domain on the rotator space such that its closure $\overline Q$ is free from strong resonances and let γ be a smooth curve on Q. Then for any small ε > 0 there exists a trajectory whose projection to Q moves in a c| log ε|^αε^1/4-neighbourhood of γ $(\alpha\ge \frac{n^2 + 2n}{{\bit r} - n - 1}$ is any constant and c = c(α) > 0) with average velocity along γ of order ε/| log ε|.