Two-dimensional shapes and lemniscates
Two-dimensional shapes and lemniscates
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二维形状和双纽带
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
H. Shapiro
中科院分区：
文献类型：
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作者：
P. Ebenfelt;D. Khavinson;H. Shapiro
平面上的形状是足够光滑的约旦曲线的等价类，其中两条曲线是等价的，如果一条曲线可以通过平移和缩放得到另一条曲线。形状的指纹是单位圆的保持方向的微分同构的等价，其中两个微分同构如果与单位盘的自同构正确组合而不同，则是等价的。指纹是通过将黎曼映射以合适的方式组合到一个形状代表的内部和外部来获得的。在本文中，我们证明了n次多项式lemmnates定义的形状与n次Blaschke积的n次根之间存在一一对应关系。lemmnates近似Hausdorff度量中的所有Jordan曲线，Blaschke积的根近似C^1范数中圆的所有保向微分同态，这一事实表明lemmnates和Blaschke积的根是形状及其指纹理论研究的自然对象。
A shape in the plane is an equivalence class of sufficiently smooth Jordan curves, where two curves are equivalent if one can be obtained from the other by a translation and a scaling. The fingerprint of a shape is an equivalence of orientation preserving diffeomorphisms of the unit circle, where two diffeomorphisms are equivalent if they differ by right composition with an automorphism of the unit disk. The fingerprint is obtained by composing Riemann maps onto the interior and exterior of a representative of a shape in a suitable way. In this paper, we show that there is a one-to-one correspondence between shapes defined by polynomial lemniscates of degree n and nth roots of Blaschke products of degree n. The facts that lemniscates approximate all Jordan curves in the Hausdorff metric and roots of Blaschke products approximate all orientation preserving diffeomorphisms of the circle in the C^1-norm suggest that lemniscates and roots of Blaschke products are natural objects to study in the theory of shapes and their fingerprints.