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

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

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)637-655
Number of pages19
JournalJournal of Difference Equations and Applications
Volume22
Issue number5
DOIs
StatePublished - May 3 2016

Bibliographical note

Funding Information:
RPM support from NSF [grant number DMS-0940366].

Keywords

  • Quadratic maps
  • critical sets
  • geometric equivalence
  • maps of the real plane
  • singularities

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