Abstract:
 It is well known that critical sets and their images play important roles in the dynamics of noninvertible maps. The critical sets for planar quadratic maps are always conic sections. It is interesting to observe that the images of these critical sets have little flexibility. For example the image of a critical set that is an ellipse is always a closed curve with three cusps. When the critical set is a hyperbola, the image of one branch is a smooth curve, while the image of the other branch is a curve with a single cusp. Other cases include critical sets which are the empty set, a single point, a single line, a parabola, two parallel lines, two intersecting lines, or the whole plane are all described. We will prove all the descriptions in each case and illustrate the geometry of how the maps in each of these cases corresponds to folding and stretching of the plane.
