Hausdorff和Mendes France的尺寸。比较研究
The dimensions of Hausdorff and Mendes France. A comparative study
摘要：
本文对平面上两类简单曲线进行了比较研究。一方面，我们得到了单位区间上的分形曲线，它们具有自相似的结构，并伴随着一个豪斯多夫维数。另一方面，我们有相反的：一类局部可求长的无界曲线，它有另一个由M定义的“分数维”。法国门德斯。我们提出了一个几何建设性的过程，将使我们能够获得-作为一系列的多边形曲线的限制-一条曲线的第一家庭，由压缩变换;和另一个第二家庭，由膨胀变换。由于这一过程的连接曲线从两个家庭，我们能够比较他们的尺寸-我们在这项工作中的目的-并获得有趣的结果，如平等的情况下，后者严格的自相似性。
英文摘要：
This paper contains a comparative study of two families of simple curves drawn in the plane. On the one hand, we have the fractal curves on the unit interval, with self-similar structure, which have associated a Hausdorff dimension. On the other hand, we have the opposite: a class of locally rectifiable unbounded curves, which have another "fractional dimension" defined by M. Mendes France. We propose a geometrical constructive process that will allow us to obtain - as the limit of a sequence of polygonal curves - one curve of the first family, by contractive transformations; and another of the second family, by expansive transformations. Thanks to this process of linking curves from both families, we are able to compare their dimensions - our aim in this work -, and to obtain interesting results such as the equality of the latter in the case of strict self-similarity.
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