Plane Maps with denominator. Part III: Nonsimple Focal Points and Related bifurcations

This paper continues the study of the global dynamic properties specific to maps of the plane characterized by the presence of a denominator that vanishes in a one-dimensional submanifold. After two previous papers by the same authors, where the definitions of new kinds of singularities, called focal points and prefocal sets, are given, as well as the particular structures of the basins and the global bifurcations related to the presence of such singularities, this third paper is devoted to the analysis of nonsimple focal points, and the bifurcations associated with them. We prove the existence of a one-to-one relation between the points of a prefocal curve and arcs through the focal point having all the same tangent but different curvatures. In the case of nonsimple focal points, such a relation replaces the one-to-one correspondence between the slopes of arcs through a focal point and the points along the associated prefocal curve that have been proved and extensively discussed in the previous papers. Moreover, when dealing with noninvertible maps, other kinds of relations can be obtained in the presence of nonsimple focal points or prefocal curves, and some of them are associated with qualitative changes of the critical sets, i.e. with the structure of the Riemann foliation of the plane.