Integrable cluster dynamics of directed networks and pentagram maps

TL;DR

This paper generalizes the integrable pentagram map using cluster algebra and Poisson geometry, providing new geometric interpretations and expanding understanding of discrete integrable systems.

Contribution

It introduces a broad family of discrete integrable maps that include the pentagram map, utilizing cluster transformations and Poisson structures.

Findings

Generalized pentagram map as a cluster transformation

Established geometric interpretations of new integrable maps

Connected directed networks with integrable discrete dynamics

Abstract

The pentagram map was introduced by R. Schwartz more than 20 years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville complete integrability of this discrete dynamical system. In 2011, M. Glick interpreted the pentagram map as a sequence of cluster transformations associated with a special quiver. Using compatible Poisson structures in cluster algebras and Poisson geometry of directed networks on surfaces, we generalize Glick's construction to include the pentagram map into a family of discrete integrable maps and we give these maps geometric interpretations. This paper expands on our research announcement arXiv:1110.0472