Non-integrability vs. integrability in pentagram maps

TL;DR

This paper investigates the integrability properties of various higher-dimensional pentagram maps, showing that many are non-integrable and proposing a broader framework for understanding their discrete integrability.

Contribution

It introduces a universal class of pentagram maps with projective duality and provides numerical evidence and conjectures on their non-integrability.

Findings

Many higher-dimensional pentagram maps are non-integrable.

A universal class of pentagram maps with projective duality is defined.

Numerical evidence suggests certain generalizations are non-integrable.

Abstract

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.