The dimensions of Hausdorff and Mendes France. A comparative study
R. Hansen, M. Piacquadio

TL;DR
This paper compares the Hausdorff dimension of fractal curves with Mendes France's fractional dimension of rectifiable curves, using a constructive process to analyze their dimensional properties and relationships.
Contribution
It introduces a geometric process linking self-similar fractal curves and rectifiable curves to compare their dimensions, revealing conditions for their equality.
Findings
Hausdorff and Mendes France dimensions can be linked via a constructive process.
Equality of the two dimensions occurs in strictly self-similar cases.
The study provides a method to compare different fractional dimensions of curves.
Abstract
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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Taxonomy
TopicsMathematical Dynamics and Fractals · Theoretical and Computational Physics · Advanced Numerical Analysis Techniques
