Pentagram Rigidity for Centrally Symmetric Octagons

TL;DR

This paper proves a special case of a conjecture linking deep diagonal pentagram maps and Poncelet polygons, demonstrating integrability for centrally symmetric octagons and analyzing the associated Lagrangian foliation.

Contribution

It establishes Arnold-Liouville integrability of the 3-diagonal pentagram map on centrally symmetric octagons, advancing understanding beyond elliptic curve cases.

Findings

The 3-diagonal map is Arnold-Liouville integrable for centrally symmetric octagons.

Detailed analysis of the Lagrangian surface foliation associated with the map.

Progress towards the conjecture connecting pentagram maps and Poncelet polygons.

Abstract

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.