Quasi-attractors in a piece-wise smooth area-preserving map

In a two-dimensional area-preserving map we found a kind of noninvertibilit y that is induced by a piece-wise smooth property of the map. This can lead to the appearance of such kinds of elliptic islands that attract the itera tions from a set of initial values outside themselves, while behaving regul arly inside. We suggest calling such islands quasi-attractors.