Transport in perturbed integrable Hamiltonian systems and the fractality of phase space

We study transport in a model perturbed integrable Hamiltonian system by calculating the volume, V(t), of elementary phase space cells visited by a trajectory, as a function of time. We use this function in order to "measure" the fractality of phase space. We argue that the "degree" of fractality is related to the well known difficulties in assigning unambiguously Lyapunov Characteristic Numbers (LCN's) to trajectories. Moreover we show that transport in phase space regions with pronounced fractality cannot be described as "normal diffusion", since the self-similar properties of V(t) imply that it is governed by Levy statistics, while the correlation dimension of $dV/dt$ implies that, in some cases, the process is strongly non-Markovian.