Quantum Hamiltonians with quasi-ballistic dynamics and point spectrum

Consider the family of Schrödinger operators (and also its Dirac version) on ℓ2(Z)ℓ2(Z) or ℓ2(N)ℓ2(N)Hω,SW=Δ+λF(Snω)+W,ω∈Ω, where S is a transformation on (compact metric) Ω, F is a real Lipschitz function and W   is a (sufficiently fast) power-decaying perturbation. Under certain conditions it is shown that Hω,SW presents quasi-ballistic dynamics for ω   in a dense GδGδ set. Applications include potentials generated by rotations of the torus with analytic condition on F, doubling map, Axiom A dynamical systems and the Anderson model. If W   is a rank one perturbation, examples of Hω,SW with quasi-ballistic dynamics and point spectrum are also presented.