On the integrability of the shift map on twisted pentagram spirals

TL;DR

This paper demonstrates the integrability of the shift map on twisted pentagram spirals by establishing a non-standard Lax representation and showing invariance of the monodromy and associated Riemann surface.

Contribution

It introduces a coordinate system and a non-standard Lax representation for the shift map on twisted spirals, proving its integrability.

Findings

The shift map has a preserved monodromy conjugation class.

The map is invariant under a 1-parameter group action.

Invariants of the shift map are generated from the monodromy.

Abstract

In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type $(N, 1)$ possesses a non-standard Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of twisted spirals, writing the map in terms of the coordinates and associating a natural parameter-free non-standard Lax representation. We then show that the map is invariant under the action of a $1$-parameter group on the moduli space of twisted $(N,1)$ spirals, which allows us to construct the Lax pair. We also show that the monodromy defines an associated Riemann surface that is preserved by the map. We use this fact to generate invariants of the shift map.