The Pentagram Map: Combinatorial and Geometric Perspectives.

[摘要] The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its shortest diagonals, and output the smaller polygon which they cut out.Expressed in suitable coordinates, this operation defines a rational map on the configuration space of polygons in the plane.We employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map.We next apply our formulas to study the singularities of the pentagram map.Singularities arise when the input polygon has certain degeneracies preventing the pentagram map from being carried out.We show that a ;;typical;; singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz.In addition, we provide a method to bypass such a singular patch by geometrically constructing the first subsequent iterate that is well-defined on the singular locus under consideration.

[发布机构] University of Michigan

[效力级别] Mathematics

[关键词] Pentagram Map;Mathematics;Science;Mathematics