Integrability of the Pentagram Map
Integrability of the Pentagram Map
复制标题
五角星图的可积性
DOI：
10.1215/00127094-2382228
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发表时间：
2011
影响因子：
2.5
通讯作者：
F. Soloviev
中科院分区：
文献类型：
--
作者：
F. Soloviev
五角星形地图是由R. Schwartz在1992年提出了凸平面多边形。最近，V. Ovsienko，R. Schwartz和S. Tabachnikov证明了刘维积分的五角星地图一般monodromies通过提供一个泊松结构和足够数量的积分对合空间的扭曲多边形。在本文中，我们证明了代数几何可积的任何monodromy，即，对于扭曲多边形和闭合多边形。为此，我们证明了五角星形映射可以写成一个离散的零曲率方程与谱参数，研究相应的谱曲线，其雅可比矩阵的动力学。我们还证明了辛叶泊松括号发现扭曲多边形的辛结构从Krichever-Phong的普遍公式相吻合。
The pentagram map was introduced by R. Schwartz in 1992 for convex planar polygons. Recently, V. Ovsienko, R. Schwartz, and S. Tabachnikov proved Liouville integrability of the pentagram map for generic monodromies by providing a Poisson structure and the sufficient number of integrals in involution on the space of twisted polygons. In this paper we prove algebraic-geometric integrability for any monodromy, i.e., for both twisted and closed polygons. For that purpose we show that the pentagram map can be written as a discrete zero-curvature equation with a spectral parameter, study the corresponding spectral curve, and the dynamics on its Jacobian. We also prove that on the symplectic leaves Poisson brackets discovered for twisted polygons coincide with the symplectic structure obtained from Krichever-Phong's universal formula.