Proceedings of the International Geometry Center

ISSN-print: 2072-9812
ISSN-online: 2409-8906
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On fractal properties of Weierstrass-type functions

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Claire David
https://orcid.org/0000-0002-4729-0733

Abstract

In the sequel, starting from the classical Weierstrass function defined, for any real number $x$, by $ {\mathcal W}(x)=\displaystyle \sum_{n=0}^{+\infty} \lambda^n\,\cos \left(2\, \pi\,N_b^n\,x \right)$, where $\lambda$ and $N_b$ are two real numbers such that~\mbox{$0 <\lambda<1$},~\mbox{$ N_b\,\in\,\N$} and $ \lambda\,N_b > 1 $, we highlight intrinsic properties of curious maps which happen to constitute a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions.

Keywords:
Weierstrass function; non-differentiability; iterative function systems

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How to Cite
David, C. (2019). On fractal properties of Weierstrass-type functions. Proceedings of the International Geometry Center, 12(2), 43–61. https://doi.org/10.15673/tmgc.v12i2.1485
Section
Papers
Author Biography

Claire David, Sorbonne University

LABORATOIRE JACQUES-LOUIS LIONS

References

1. Krzysztof Baranski, Balazs Barany, Julia Romanowska. On the dimension of the graph of the classical Weierstrass function. Adv. Math., 265:32-59, 2014, \\printDOI10.1016/j.aim.2014.07.033.,
2. M. F. Barnsley, S. Demko. Iterated function systems and the global construction of fractals. Proc. Roy. Soc. London Ser. A, 399(1817):243-275, 1985, \\printDOI10.1098/rspa.1985.0057.,
3. A. S. Besicovitch, H. D. Ursell. Sets of fractional dimensions (V): on dimensional numbers of some continuous curves. J. London Math. Soc., s1-12(1):18-25, 1937, \\printDOI10.1112/jlms/s1-12.45.18.,
4. Claire David. Bypassing dynamical systems: a simple way to get the box-counting dimension of the graph of the Weierstrass function. Proc. Int. Geom. Cent., 11(2):53-68, 2018, \\printDOI10.15673/tmgc.v11i2.1028.,
5. Claire David. Wandering across the Weierstrass function, while revisiting its properties. To appear, 2019.,
6. Robert L. Devaney. An introduction to chaotic dynamical systems. Studies in Nonlinearity. Westview Press, Boulder, CO, 2003. Reprint of the second (1989) edition.,
7. Brian R. Hunt. The Hausdorff dimension of graphs of Weierstrass functions. Proc. Amer. Math. Soc., 126(3):791-800, 1998, \\printDOI10.1090/S0002-9939-98-04387-1.,
8. John E. Hutchinson. Fractals and self-similarity. Indiana Univ. Math. J., 30(5):713-747, 1981, \\printDOI10.1512/iumj.1981.30.30055.,
9. Gerhard Keller. A simpler proof for the dimension of the graph of the classical Weierstrass function. Ann. Inst. Henri Poincare Probab. Stat., 53(1):169-181, 2017, \\printDOI10.1214/15-AIHP711.,
10. Jun Kigami. A harmonic calculus on the Sierpinski spaces. Japan J. Appl. Math., 6(2):259-290, 1989, \\printDOI10.1007/BF03167882.,
11. Benoit B. Mandelbrot. Fractals: form, chance, and dimension. W. H. Freeman and Co., San Francisco, Calif., revised edition, 1977. Translated from the French.,
12. Benoit B. Mandelbrot. The fractal geometry of nature. W. H. Freeman and Co., San Francisco, Calif., 1982. Schriftenreihe fur den Referenten.,
13. K. Weierstrass. Uber kontinuierliche Funktionen eines reellen arguments, die fur keinen Wert des letzteren einen bestimmten Differentialquotienten besitzen. Journal fur die reine und angewandte Mathematik, 79:29-31, 1875.