International Journal of Bifurcation and Chaos, Vol. 13, No. 8 (2003) 2253–2277 c  World Scientific Publishing Company PLANE MAPS WITH DENOMINATOR. PART II: NONINVERTIBLE MAPS WITH SIMPLE FOCAL POINTS GIAN-ITALO BISCHI ∗ and LAURA GARDINI † Istituto di Scienze Economiche, University of Urbino, Via Saffi, 45, 61029 PU, Italy ∗ bischi@econ.uniurb.it † gardini@econ.uniurb.it CHRISTIAN MIRA 19 rue d’Occitanie, 31130 Quint Fonsegrives, France c.mira@free.fr Received April 18, 2002; Revised June 5, 2002 This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves.A prefocal curve is a set of points which are mapped (or “focalized”) into a single point, called focal point, by the inverse map when it is invertible, or by at least one of the inverses when it is noninvertible. In the case of noninvertible maps, the previous text dealt with the simplest geometrical situation, which is nongeneric. To be more precise this situation occurs when several focal points are associated with a given prefocal curve. The present paper defines the generic case for which only one focal point is associated with a given prefocal curve. This is due to the fact that only one inverse of the map has the property of focalization, but with properties different from those of invertible maps. Then the noninvertible maps of the previous Part I appear as resulting from a bifurcation leading to the merging of two prefocal curves, without merging of two focal points. Keywords : Noninvertible maps; vanishing denominator; focal points; prefocal sets. 1. Introduction In a previous paper [Bischi et al., 1999] (Part I henceforth), we studied some global dynamical properties of two-dimensional maps T , related to the presence of a denominator which vanishes in a one-dimensional subset of the plane. Differently from the present paper, which is limited to the case of noninvertible maps, in Part I we considered both invertible and noninvertible maps. We remind that the phase space of a noninvertible map is subdi- vided into open regions (or zones) Z k , whose points have k distinct rank-1 preimages, obtained by the application of k distinct inverses of the map. A spe- cific feature of noninvertible maps is the existence of the critical set LC (from “Lignes Critique”, see [Gumowski & Mira, 1980; Mira et al., 1996]) defined as the locus of points having at least two coinci- dent rank-1 preimages, located on the set of merg- ing preimages denoted by LC −1 , LC = T (LC −1 ). Segments of LC are boundaries that separate dif- ferent regions Z k , but the converse is not generally true, that is, also in the two-dimensional case, as in the one-dimensional one, boundaries of regions Z k which are not portions of LC may exist. 2253