Quantum Hamiltonians with quasi-ballistic dynamics and point spectrum

Consider the family of Schrödinger operators (and also its Dirac version) on or 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 presents quasi-ballistic dynamics for ω in a dense Gδ 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 with quasi-ballistic dynamics and point spectrum are also presented.