Pentagram Maps on Coupled Polygons: Integrability, Geometry and Orthogonality

In this paper, a family of novel generalizations of the (dented) pentagram map are proposed on the space of coupled polygons. Their geometry and integrability are described together with some explicit examples. In particular, we find that these maps are related to refactorization maps in the Poisson–Lie group of pseudo-difference operators associated with pairs of non-disjoint progressions \(\textrm{J}_{\pm }\subset {\mathbb {Z}}\) ${\text{J}}_{\pm }\subset \mathbb{Z}$ , which resolves an open question proposed by Izosimov. Furthermore, it is shown that these maps can be associated with a class of bi-orthogonal polynomials, which generalize the corresponding result for the classical Laurent bi-orthogonal polynomials.