INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS
INTEGRABLE CLUSTER DYNAMICS OF DIRECTED NETWORKS AND PENTAGRAM MAPS
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有向网络和五角星图的可积簇动力学
DOI：
10.1016/j.aim.2016.03.023
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发表时间：
2014
影响因子：
1.7
通讯作者：
A. Vainshtein
中科院分区：
文献类型：
--
作者：
M. Gekhtman;M. Shapiro;S. Tabachnikov;A. Vainshtein
五角星形地图是由R. 20多年前的施瓦茨。2009年，V. Ovsienko，R. Schwartz和S. Tabachnikov建立了这个离散动力系统的Liouville完全可积性。2011年，M.格利克将五角星形地图解释为与特殊的“星座”相关的一系列簇变换。利用Poisson和簇结构的相容性以及曲面上有向网络的Poisson几何，我们推广了Glick的构造，将五角星形映射包含到离散可积映射族中，并给出了这些映射的几何解释.附录将这些离散映射中最简单的映射与户田晶格及其三哈密顿结构联系起来。
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.