2000/04/10 by R. O. Hansen, R. Hansen, Hansen, R. +2
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Theoretical and Computational Physics #math-ph #math.CA #math.MG #math.MP
paper · pdf · doi:10.48550/arxiv.math/0004060
14 pages
arxiv created 2000/04/10 · openalex publication_date 2000/04/10 · arxiv updated 2015/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.