The Abundance of Strange Attractor in the Families of Nearby Singular Diffeomorphism

In this paper, we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space, such that for each point of the set the corresponding diffeomorphism possesses strange attractor. For some families of one-dimensional mapping satisfying certain transversality condition, we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor. Furthermore, we study the dynamics of this type of strange attractor.