Liouville–Arnold integrability of the pentagram map on closed polygons
Liouville–Arnold integrability of the pentagram map on closed polygons
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闭合多边形上五角星图的刘维尔-阿诺德可积性
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
S. Tabachnikov
中科院分区：
文献类型：
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作者：
V. Ovsienko;R. Schwartz;S. Tabachnikov
五角形映射是在投影平面上多边形的模空间上定义的离散动力系统。这张图最近引起了相当大的兴趣，主要是因为它与许多不同领域的联系，如经典射影几何、代数组合、模空间、聚类代数和可积系统。五角形映射的可积性是由Schwartz猜想出来的，本文作者证明了五角形映射在更大的扭曲多边形空间中的可积性。本文证明了五角形映射在闭多边形模空间上是完全可积的初始猜想。对于实射影平面上的凸多边形，这一结果暗示了模空间上环面叶理的存在。叶理的叶片具有仿射结构，五角星图的动力学是准周期的。我们的证明是基于扭曲多边形空间上的不变泊松结构。证明了对应于单不变量的哈密顿向量场保持了闭多边形的空间，并在单不变量的水平面上定义了一个不变的仿射结构。
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.