School of Mathematics
University of Minnesota
Quadratic Cusp


Classification of critical sets and their images for quadratic maps of the plane

Chia-Hsing Nien, Bruce B. Peckham, and Richard P. McGehee

Journal of Difference Equations and Applications 22 (5) (2016) 637-655. doi: 10.1080/10236198.2015.1127360


Abstract. We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose critical sets are either ellipses or hyperbolas, was published by Delgado et al. in 2013. This work completes the classification by including all the nongeneric cases: the empty set, a single point, a single line, a parabola, two parallel lines, two intersecting lines, or the whole plane. We describe all possible images for each critical set case and illustrate the geometry of representative maps for each case.

Electronic Copies

The paper is available for subscribers on the publisher's Web site:
doi: 10.1080/10236198.2015.1127360

A preprint was posted on ArXiv on 9 July 2015:
Cached copy: Nien2015ArXiv1507.pdf


Last update: May 13, 2018 ©2018 Richard McGehee