Speaker: Stéphane Seuret, Université Paris-Est Créteil

Time : 16:00 - 17.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

YouTube : https://youtu.be/8g-LbyKaCB0

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Fractals, multifractals, dynamical systems, invariant measures, Hausdorff dimension

Abstract: In a famous paper published in 1904 [1, 2], 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.

This is a joint work with Zoltan Buczolich (Eotvos Lorand University) and Yann Demichel (Universite Paris Nanterre)

Biography: Stephane Seuret is Full professor in Mathematics at Universite Paris Est Creteil (UPEC). He is an internationaly recognized expert in fractal geometry, multifractals, and their connexion with probability theory, dynamical systems and metric number theory. He was president of the French Mathematical Society (2016-2020) and is now head of the ”Laboratoire d’Analyse et de Mathematiques Appliquees” (LAMA), which gathers 120 mathematicians at UPEC.

Bibliography

[1] H. von Koch, Sur une courbe continue sans tangente, obtenue par une construction geometrique elementaire, Arkiv f¨or matematik astronomi och fysik, 1 681-702 (1904). Reprinted in English in Classics on Fractals, G. A. Edgar, Addison-Wesley Publishing (1993) 25–45. https://staff.math.su.se/lenb/dok/von-Koch-1904.pdf
[2] H. von Koch, Une methode geometrique elementaire pour l’etude de certaines questions de la th´eorie des courbes planes, Acta Mathematica, 1 99-04 145–174