The multifractal nature of a parametrized family of von Koch functions
Abstract
In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes. In fact, von Koch's main goal was the construction of a continuous but nowhere differentiable function, very similar to the snowflake, using elementary geometric procedures, and not analytical formulae. We prove that a parametrized family of functions (including and) generalizing von Koch's example enjoys a rich multifractal behavior, thus enriching the class of historical mathematical objects having surprising regularity properties. The analysis relies on the study of the orbits of an underlying dynamical system and on the introduction of self-similar measures and non-trivial iterated functions systems adapted to the problem.
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BibTeX
@article{buczolich2026multifractal,
title = {{The multifractal nature of a parametrized family of von Koch functions}},
author = {Zoltán Buczolich and Yann Demichel and Stéphane Seuret},
journal = {Advances in Mathematics},
year = {2026},
month = sep,
doi = {10.1016/j.aim.2026.111157},
url = {https://hal.science/hal-04987686v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - The multifractal nature of a parametrized family of von Koch functions
AU - Zoltán Buczolich
AU - Yann Demichel
AU - Stéphane Seuret
PY - 2026
DA - 2026/09/15
JO - Advances in Mathematics
DO - 10.1016/j.aim.2026.111157
UR - https://hal.science/hal-04987686v2
AB - In a famous paper published in 1904, Helge von Koch introduced the curve that still serves nowadays as an iconic representation of fractal shapes. In fact, von Koch's main goal was the construction of a continuous but nowhere differentiable function, very similar to the snowflake, using elementary geometric procedures, and not analytical formulae. We prove that a parametrized family of functions (including and) generalizing von Koch's example enjoys a rich multifractal behavior, thus enriching the class of historical mathematical objects having surprising regularity properties. The analysis relies on the study of the orbits of an underlying dynamical system and on the introduction of self-similar measures and non-trivial iterated functions systems adapted to the problem.
SN - 0001-8708
SN - 1090-2082
ER -