闭多边形上五角星形映射的Liouville-Arnold可积性
Liouville-Arnold integrability of the pentagram map on closed polygons
摘要：
五角星映射是定义在射影平面上多边形模空间上的离散动力系统。这个映射最近吸引了相当大的兴趣，主要是因为它连接到一些不同的域，如：经典射影几何，代数组合，模空间，簇代数和可积系统。R. Schwartz和后来由V. Ovsienko，R. Schwartz和S. Tabachnikov的一个更大的空间扭曲的多边形。本文证明了闭多边形的模空间上五角星形映射完全可积的初步猜想。对于实射影平面上的凸多边形，这个结果意味着在模空间上存在复曲面叶理。叶理的叶子具有仿射结构，五角星形映射的动力学是准周期的。我们的证明是基于一个不变的泊松结构的扭曲多边形的空间。证明了对应于单值不变量的Hamilton向量场保持闭多边形空间，并在单值不变量的水平曲面上定义了一个不变的仿射结构.
英文摘要：
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 R. Schwartz and later proved by V. Ovsienko, R. Schwartz and S. Tabachnikov for a larger space of twisted polygons. In this paper, 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 quasi-periodic. 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 monodoromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.
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