Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.

In this paper we study the behavior of diffeomorphisms, contained in the closure \overline{\mathcal{A}_\alpha} (in the inductive limit topology) of the set \mathcal{A}_\alpha of real-analytic diffeomorphisms of the torus \mathbb T^2, which are conjugated to the rotation R_\alpha:(x,y)\mapsto (x + \alpha , y) by an analytic measure-preserving transformation. We show that for a generic \alpha\in [0,1], \overline{\mathcal{A}_\alpha} contains a dense set of uniquely ergodic diffeomorphisms. We also prove that \overline{\mathcal{A}_\alpha} contains a dense set of diffeomorphisms that are minimal and non-ergodic.