Liouville–Arnold integrability of the pentagram map on closed polygons
By Valentin Ovsienko, Richard Evan Schwartz, Serge Tabachnikov. 2011
73 citations recorded by the source corpus.
The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems. Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/1d468b85073edc5fedb33040b34ea29ae052381c
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
- Sophie Morier-Genoud
- Valentin Ovsienko
- Serge Tabachnikov
- Max Glick
- J. '. Paiva
- G. Berck
- Richard Evan Schwartz
In Statistical and Nonlinear Physics
Citations in the corpus
- 2-frieze patterns and the cluster structure of the space of polygons (2010, methodology): “It is not clear, at the time of writing, how the cluster structure on the space C n that we describe in this paper is related to complete integrability of the map T on C n proved in [28] and [21].”
- INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS (2014, unknown): “His approach established complete integrability of the pentagram map on the space of closed polygons Cn as well; a different proof of this result was given in [33].”