Plane maps with denominator. I. Some generic properties

This paper is devoted to the study of some global dynamical properties and bifurcations of two-dimensional maps related to the presence, in the map or in one of its inverses, of a vanishing denominator. The new concepts of fo cal points and of prefocal curves are introduced in order to characterize s ome new kinds of contact bifurcations specific to maps with denominator. Th e occurrence of such bifurcations gives rise to new dynamic phenomena, and new structures of basin boundaries and invariant sets, whose presence can o nly be observed if a map (or some of its inverses) has a vanishing denomina tor.