五角星映射的拟周期运动
Quasiperiodic Motion for the Pentagram Map
摘要：
五角星形映射是定义在多边形上的投影自然迭代，也是定义在广义的多边形概念上的投影自然迭代，我们称之为{ \it扭曲多边形}。本文描述了我们最近在五角星形映射上的工作，发现了扭曲多边形空间上的Poisson结构，并证明了相对于这种Poisson结构的五角星形映射在Arnold-Liouville意义下是完全可积的.对于某些扭曲的多边形，如那些我们称之为{ \it普遍凸}，我们翻译成一个关于五角形映射轨道的准周期概念的陈述的可积性。我们还解释了如何连续极限的五角星映射是经典的Boissinesq方程，一个完全可积的偏微分方程。
英文摘要：
The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call {\it twisted polygons}. In this note we describe our recent work on the pentagram map, in which we find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call {\it universally convex}, we translate the integrability into a statement about the quasi-periodic notion of the pentagram-map orbits. We also explain how the continuous limit of the Pentagram map is the classical Boissinesq equation, a completely integrable PDE.
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