[Certaines propriétés géométriques de la fonction non différentiable de Riemann]
La célèbre fonction non différentiable de Riemann est une fonction continue, mais presque nulle part dérivable. Des simulations numériques montrent qu'une de ses versions complexes représente une trajectoire temporelle dans le cadre de l'équation du flot binormal, aussi connue sous le nom de Vortex Filament Equation. Par conséquent, on analyse certaines propriétés géométriques de son image dans . Dans cette note, on affirme que la dimension de Hausdorff de l'image n'est jamais plus grande que 4/3 et qu'elle n'a pas de tangentes.
Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in . The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.
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@article{CRMATH_2019__357_11-12_846_0,
author = {Eceizabarrena, Daniel},
title = {Some geometric properties of {Riemann's} non-differentiable function},
journal = {Comptes Rendus. Math\'ematique},
pages = {846--850},
publisher = {Elsevier},
volume = {357},
number = {11-12},
year = {2019},
doi = {10.1016/j.crma.2019.10.007},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2019.10.007/}
}
TY - JOUR AU - Eceizabarrena, Daniel TI - Some geometric properties of Riemann's non-differentiable function JO - Comptes Rendus. Mathématique PY - 2019 SP - 846 EP - 850 VL - 357 IS - 11-12 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2019.10.007/ DO - 10.1016/j.crma.2019.10.007 LA - en ID - CRMATH_2019__357_11-12_846_0 ER -
%0 Journal Article %A Eceizabarrena, Daniel %T Some geometric properties of Riemann's non-differentiable function %J Comptes Rendus. Mathématique %D 2019 %P 846-850 %V 357 %N 11-12 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2019.10.007/ %R 10.1016/j.crma.2019.10.007 %G en %F CRMATH_2019__357_11-12_846_0
Eceizabarrena, Daniel. Some geometric properties of Riemann's non-differentiable function. Comptes Rendus. Mathématique, Tome 357 (2019) no. 11-12, pp. 846-850. doi : 10.1016/j.crma.2019.10.007. http://www.numdam.org/articles/10.1016/j.crma.2019.10.007/
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