Iterates of Mappings Contracting Perimeters of Triangles

Iterates of mappings contracting perimeters without a periodic point of a prime period two are graphic contractions except for finitely many. The reverse conclusion does not hold, there are graphic contractions without iterates that are mappings contracting perimeters of triangles. We prove that a mapping contracting perimeters of triangles cannot have a periodic point of a prime period greater than two, and prove the existence of a periodic point using the Saturated principle of graphic contraction. The theory is substantiated with numerous examples.