Anisotropic properties of riddled basins

We consider some general properties of chaotic attractors with riddled basins of attraction (basins with positive measure but open dense complements) in dynamical systems with symmetries. We investigate how a basin of attraction can be riddled in some directions and not in others if the attractor is contained in the intersection of several invariant subspaces. This means that the extreme sensitivity to added noise (bubbling) associated with riddled basin attractors is in fact strongly dependent on the nature of the noise. We discuss examples of this for systems of globally coupled maps.