Integrable cluster dynamics of directed networks and pentagram maps

Michael Gekhtman, Michael Shapiro, Serge Tabachnikov, Alek Vainshtein

Research output: Contribution to journalArticlepeer-review


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 compatibility of Poisson and cluster structures 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. The appendix relates the simplest of these discrete maps to the Toda lattice and its tri-Hamiltonian structure.

Original languageEnglish
Pages (from-to)390-450
Number of pages61
JournalAdvances in Mathematics
StatePublished - 10 Sep 2016

Bibliographical note

Publisher Copyright:
© 2016 Elsevier Inc.


  • Cluster dynamics
  • Generalized pentagram maps
  • Liouville integrability
  • Mutations
  • Networks on surfaces

ASJC Scopus subject areas

  • General Mathematics


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