Diffusion in three-dimensional liouvillian maps

It is shown that chaotic trajectories in volume-preserving flows, rI=u(x,y,z,t), which are arbitrarily close to integrability, 0<1, can be either trapped or diffusive throughout the available space. A classification of these flows is proposed which both distinguishes and predicts the appropriate type of behavior. In the unbounded case, a new mechanism of diffusion is found which combines motion on the resonances with an adiabatic drift. This process is reminiscent of Arnold diffusion.