Chaos and fractal properties induced by noninvertibility of models in the form of maps

Many classes of discrete dynamical systems give rise to models in the form of noninvertible maps. With respect to invertible maps, noninvertible maps introduce a singularity of a different nature: the critical set of rank-one, as the geometrical locus of points having at least two coincident preimages. Such new singularities play a fundamental role in the definition of attractors, basins and their bifurcations. The purpose of this paper is a survey of some fundamental results related to two-dimensional noninvertible maps leading to specific chaotic behaviors, as fractal sets, characterizing irreversibility properties of a class of discrete systems.