面积归一化五角星映射动力学：谱扁平化与椭圆渐近行为
Area-Normalized Pentagram Map Dynamics: Spectral Flattening and Elliptic Asymptotics
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中文总结 AI 辅助
该研究针对经面积归一化的五角星映射，证明了多边形动力学的谱二分性，并将框架应用于五边形、六边形及庞塞莱多边形，得出相关谱结论与几何性质。
中文摘要 AI 辅助
五角星映射将多边形映射为其连续短对角线的交点。我们研究其形状动力学，方法是对每个迭代结果进行平移和正缩放，以恢复无符号面积和重心。对于那些在经过有限次迭代和循环重标记后，动力学由单个射影变换生成的多边形，我们证明了谱二分性：若存在主导实射影直线，则会产生扁平化和无界直径；若存在主导实特征值与一对次主导非实特征值，则会在同心相似椭圆上产生渐近运动。我们将该框架应用于通过Glick算子处理的五边形和六边形，以及通过Darboux–Schwartz射影变换处理的庞塞莱多边形，得到了谱图和庞塞莱返回谱的雅可比函数公式。由此，我们恢复了Schwartz关于严格凸非射影正则五边形的细长结果，证明了在明确非退化假设下凸六边形的直线或椭圆二分性，并确立了严格凸非射影正则庞塞莱多边形的扁平化性质。
英文摘要
The pentagram map sends a polygon to the intersections of consecutive short diagonals. We study its shape dynamics after translating and positively rescaling each iterate to restore unsigned area and barycenter. For polygons whose dynamics is generated, after passing to a finite iterate and cyclic relabelling, by one projectivity, we prove a spectral dichotomy. A dominant real projective line yields flattening and unbounded diameter, whereas a dominant real eigenvalue with a subdominant non-real pair produces asymptotic motion on concentric homothetic ellipses. We apply this framework to pentagons and hexagons via Glick's operator and to Poncelet polygons via the Darboux--Schwartz projectivity. We obtain spectral diagrams and a Jacobi-function formula for the Poncelet return spectrum. Consequently, we recover Schwartz's long-and-thin result for strictly convex non-projectively-regular pentagons, prove a line-or-ellipse dichotomy for convex hexagons under explicit nondegeneracy assumptions, and establish flattening for strictly convex non-projectively-regular Poncelet polygons.
