扭曲五角星螺线上移位映射的可积性
On the integrability of the shift map on twisted pentagram spirals
摘要：
本文证明了定义在$(N, 1)$型扭五角星螺线模空间上的移位映射具有一个非标准的Lax表示，该表示伴随一个保共轭类的单值映射.我们证明了这一点，找到一个坐标系在模空间的扭曲螺旋，写地图的坐标和关联的自然参数自由的非标准的拉克斯表示。然后，我们表明，该地图是不变的作用下的$1$参数组的扭曲$(N,1)$螺旋的模空间，这使我们能够构建的Lax对。我们还表明，monodromy定义了一个相关联的黎曼曲面，保存的地图。我们使用这个事实来生成移位映射的不变量。
英文摘要：
In this paper we prove that the shift map defined on the moduli space of twisted pentagram spirals of type $(N, 1)$ possesses a non-standard Lax representation with an associated monodromy whose conjugation class is preserved by the map. We prove this by finding a coordinate system in the moduli space of twisted spirals, writing the map in terms of the coordinates and associating a natural parameter-free non-standard Lax representation. We then show that the map is invariant under the action of a $1$-parameter group on the moduli space of twisted $(N,1)$ spirals, which allows us to construct the Lax pair. We also show that the monodromy defines an associated Riemann surface that is preserved by the map. We use this fact to generate invariants of the shift map.
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