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arXiv · 1406.1883

Integrable cluster dynamics of directed networks and pentagram maps

Michael Gekhtman·Michael Shapiro·Serge Tabachnikov·Alek Vainshtein
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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

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Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein. 2014-06-07. Integrable cluster dynamics of directed networks and pentagram maps. https://arxiv.org/abs/1406.1883

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