Abstract

It has been known for more than a decade that, if a self-similar arc \(\gamma \) can be shifted along itself by similarity maps that are arbitrarily close to identity, then \(\gamma \) is a straight line segment. We extend this statement to the class of self-affine arcs and prove that each self-affine arc admitting affine shifts that may be arbitrarily close to identity is a segment of a parabola or a straight line.

中文翻译：

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自仿射弧的刚性定理

摘要

十多年来人们都知道，如果一条自相似弧\(\gamma \)可以通过任意接近身份的相似图沿着自身移动，那么\(\gamma \)就是一条直线段. 我们将此陈述扩展到自仿射弧的类别，并证明每个允许可能任意接近身份的仿射位移的自仿射弧是抛物线或直线的一段。