Abstract
The questions of fractal's constructions interest the researchers
since many years and the method of iterated functions systems is currently very
used.
Our work is in order to present the fractals and their
geometry and to introduce the method of construction based on iterated
functions systems (IFS) and in particular, those on multifunctions. To do this,
we have given in this thesis some notions on topology and on measure theory of
the Euclidien space IV necessary to the study of fractals. We have then
introduced the notions of Hausdorff dimension and that of topological dimension
to give the mathematical definition of fractals. The geometry of fractals and
some well known examples and their program in Matlab® have been also
presented.
In the last part of this thesis, we were interested in the IFS
technics and in particular those on multifunctions.
Keywords
Fractal, Hausdorff dimension, topological dimension, Hausdorff
measure, iterated functions systems, multifunctions, outer measure.
Table des matières
Remerciements iii
Résuméiii
Notations et symboles ix
Introduction xi
1
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Mesures et structures boréliennes
1.1 Mesure positive
1.2 Mesure extérieure
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1 1
5
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1.2.1 Prolongement d'une mesure
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7
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1.2.2 Exemples de mesures extérieures
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9
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2
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Dimension et mesures de Hausdorff
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11
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2.1
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Mesures de Hausdorff
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11
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2.2
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Dimension de Hausdorff
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19
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2.3
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Calculs des dimensions
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23
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3
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Dimension topologique
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33
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4
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La géométrie des fractals
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37
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4.1
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Système de fonctions itérées
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38
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4.1.1 Dimensions des ensembles auto-similaires
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47
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4.1.2 Système de fonctions itérées affines
dans R2
50
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4.2
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Système de fonctions itérées complexes
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54
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4.2.1 Les ensembles de Julia 55
4.2.2 L'ensemble de Mandelbrot 61
4.3 Systeme de fonctions it'er'ees sur les multifonctions 65
Conclusion et perspectives 71
Programmes sur les fractals 72
.1 L'ensemble de Cantor 72
.2 Courbe de von Koch 73
.3 Triangle de Sierpenski 74
.4 Fougere de Barnsley 76
.5 Poussiere de Cantor 77
.6 Tapis de Sierpenski 78
.7 L'ensemble de Julia 80
.8 L'ensemble de Mandelbort 81
R'ef'erences bibliographiques 82
Table des figures
2.1 Graphe de 7(8(F) 20
2.2 L'ensemble triadique de Cantor 23
2.3 La courbe de von Koch 25
2.4 La courbe de Koch quadratique de type 1 26
2.5 La courbe de Koch quadratique de type 2 27
2.6 Le flocon de von Koch 28
2.7 Le triangle de Sierpinski 28
2.8 Recouvrements du triangle de Sierpinski 29
2.9 Le tapis de Sierpinski 30
2.10 L'éponge de Menger 30
2.11 La courbe de Peano 31
3.1 Poussière de Cantor 36
4.1 Exemple d'auto-similarité 38
4.2 La courbe de Koch quadratique modifié 49
4.3 La fougère de Barnsley 54
4.4 Ensemble de Mandelbrot 62
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