The geometry of dented pentagram maps

TL;DR

This paper introduces a new family of higher-dimensional generalizations of the pentagram map called dented pentagram maps, proves their integrability, and explores their geometric and algebraic properties.

Contribution

It defines dented pentagram maps in higher dimensions, proves their integrability, and analyzes their geometric structure and continuous limits, extending previous pentagram map results.

Findings

Proved integrability of dented pentagram maps in 3D.

Described geometry and Lax representations of these maps.

Compared invariant tori dimensions with other pentagram maps.

Abstract

We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d-1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps with those for the higher pentagram maps constructed with the help of short diagonal hyperplanes. When restricted to corrugated polygons, the dented pentagram maps coincide between themselves and with the corresponding corrugated pentagram map. Finally, we prove integrability for a variety of pentagram maps for generic and partially corrugated polygons in higher dimensions.