On Integrable Generalizations of the Pentagram Map

Abstract

In this paper we prove that the generalization to RP n of the pentagram map defined in [3] is invariant under certain scalings for any n. This property allows the definition of a Lax representation for the map, to be used to establish its integrability. arXiv:1303.4295v1 [math.DS]

Open problems, questions and challenges in finite- dimensional integrable systems

“…For some of these maps, for example, T (2,2),(1,1) , T (2,1), (1,1) , and T (3,1), (1,1) , complete integrability has been established [131,133,156]. Other cases were studied numerically in [132].…”

The Limit Point of the Pentagram Map

“…The pentagram map has seen a spike in popularity in the current decade thanks largely to the discovery that it is a discrete integrable system [1,9,10,14], and also because of emerging connections with cluster algebras [1,2]. In a sense, the recent work differs significantly from the first paper [11] in that (1) for the purposes of integrability and cluster algebras, it is more natural to have the pentagram map act not on individual polygons but on projective equivalence classes of polygons, and (2) there has been a focus on generalized pentagram maps [3][4][5][6][7][8], which are not known to possess a property analogous to preserving convexity. The present paper returns to the matter of the limit point (X, Y ) of the pentagram map acting on a convex polygon A.…”

The pentagram integrals for Poncelet families

“…In [MB1], Mari-Beffa defines higher dimensional generalizations of the pentagram map and relates their continuous limits to various families of integrable PDEs. See also [MB2].…”