Attraction basins in discretized maps

In this note we consider maps which are defined on continuous space whose large time behaviour displays a strange attractor. We are interested in the properties of the discrete maps that are obtained from these continuous ones by discretizing the space. Such systems behave as disordered dynamical systems. The strange attractor breaks down in many (sometimes one) periodic attractors. We study here the statistical properties of such attractors. Generalizing previous conjectures we propose that the distribution of the attraction basins' sizes is the same as in the random map problem. This result is shown to be in good agreement with numerical experiments.