×

Self-similar structure on intersections of triadic Cantor sets. (English) Zbl 1151.28009

For a triadic Cantor set \(\mathbf C\) the authors investigate the intersections \(\mathbf C\cap (\mathbf C+ \alpha)\). All real numbers \(\alpha\) are characterized for which \(\mathbf C\cap (\mathbf C+ \alpha)\) is a self-similar set, and the form and structure of corresponding iterated function systems are studied.
An iterated function system for a nonempty compact set \(T\) is defined as a family of distinct functions \(\{f_i(x)=r_ix+b_i,0<|r_i|<1\}_{i=1}^N\) such that \(T={\bigcup_{i=1}^N f_i(T)};\) \(T\) is called a self-similar set. The set of all iterated function systems for \(T\) is denoted by \(F(T).\) The classical Cantor triadic set \(\mathbf C\) is self-similar, and the simplest iterated function system for it is \(\{f_1(x)=x/3\), \(f_2(x)=x/3+2/3\}.\) The paper studies two principal questions on Cantor sets, namely: (1) Whether or not an intersection \(\mathbf C\cap (\mathbf C+ \alpha)\) of the Cantor set and its translation is a self-similar set? (2) If (1) has a confirmative answer, what is the form of an iterated function system for it?
The authors show that when studying the self-similarity of the set \(\mathbf C\cap (\mathbf C+ \alpha),\) it is sufficient to consider \(\alpha\in(0,1);\) in this case, for \(\alpha=\sum_{i=1}^\infty\alpha_i3^{-i},\) \(\alpha_i\in\{-2, 0, 2\},\) they prove that \(\mathbf C\cap (\mathbf C+ \alpha)\) is a self-similar set if and only if \(\widehat\alpha\) is periodic where \(\widehat\alpha_i = 2-|\alpha_i|\) for all \(i\geq 1.\) Furthemore, if \(\mathbf C\cap (\mathbf C+ \alpha)\) is a self-similar set with more than one point, then there exists an iterated function system for it satisfying the strong separation condition.
Then they derive the general form of an iterated function system for the self-similar set \({\mathbf C}_\alpha,\) which is a translation of \(\mathbf C\cap (\mathbf C+ \alpha)\) such that the origin is the left end of it: Let \(\widehat\alpha=I\overline{(I+J)}\) with \(I, J\in\{0,2\}^p.\) Then any iterated function system \(\{f_i(x)=r_ix+b_i,0<|r_i|<1\}_{i=1}^N\) for \({\mathbf C}_\alpha\) satisfies that \(r_i=3^{-q_i}\) for some integer \(q_i\) and \(b_i=\sum_{k=1}^{p+q_i}b_{ik}3^{-k},\) \(i=1,2,\dots , N,\) where all \(b_{ik}=0\) or 2. Moreover, each \(q_i\) is a period of \(\widehat\alpha.\)
In the following the structure of all iterated function systems for the self-similar set \({\mathbf C}_\alpha\) is investigated further. For \(\widehat\alpha=I\overline{(I+J)}\) with the smallest period \(p_0,\) where again \(I, J\in\{0,2\}^p,\) it is proved that the set \(F({\mathbf C}_\alpha)\) of all iterated function systems for \({\mathbf C}_\alpha\) has a generating element if and only if \(\widehat\alpha_k\leq\widehat\alpha_{k+p_0}\) for each \(k\) (\(1\leq k\leq p+p_0\)).
Another theorem proves that if \(F({\mathbf C}_\alpha)\) has a generating element \(\mathcal G,\) then the iterated function system \(\mathcal F\) for \({\mathbf C}_\alpha\) satisfies the strong separation condition if and only if \(\mathcal F\) is a tight set in the tree of \(\mathcal G.\)
The paper is concluded by an example illustrating the complicated structure of all iterated function systems which generate \({\mathbf C}_\alpha\) when a generating element does not exist.

MSC:

28A80 Fractals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Allain, C. C.; Cloitre, M., Characterizing lacunarity of random and deterministic fractal sets, Phys. Rev. A, 44, 3552-3558 (1991)
[2] Barnsley, M.; Barnsley, L., Fractal image compression, (Image Processing: Mathematical Methods and Applications. Image Processing: Mathematical Methods and Applications, Cranfield, 1994. Image Processing: Mathematical Methods and Applications. Image Processing: Mathematical Methods and Applications, Cranfield, 1994, Inst. Math. Appl. Conf. Ser. New Ser., vol. 61 (1997), Oxford Univ. Press: Oxford Univ. Press Oxford), 183-210 · Zbl 1114.68568
[3] Davis, G. J.; Hu, T.-Y., On the structure of the intersection of two middle third Cantor sets, Publ. Mat., 39, 1, 43-60 (1995) · Zbl 0841.28001
[4] G.-T. Deng, X.-G. He, Self-similarity of unions of Cantor sets, in press; G.-T. Deng, X.-G. He, Self-similarity of unions of Cantor sets, in press
[5] Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications (1990), John Wiley & Sons · Zbl 0689.28003
[6] Falconer, K. J., Techniques in Fractal Geometry (1997), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0869.28003
[7] D.-J. Feng, Y. Wang, On the structures of generating iterated function systems of Cantor sets, in press; D.-J. Feng, Y. Wang, On the structures of generating iterated function systems of Cantor sets, in press
[8] Hutchinson, J. E., Fractal and self similarity, Indiana Univ. Math. J., 30, 713-747 (1981) · Zbl 0598.28011
[9] Lagarias, J. C.; Wang, Y., Integral self-affine tiles in \(R^n\), I. Standard and non-standard digit sets, J. London Math. Soc., 54, 161-179 (1996) · Zbl 0893.52014
[10] Li, W.-X.; Xiao, D.-M., Intersection of translations of Cantor triadic set, Acta Math. Sci., 19, 214-219 (1999) · Zbl 0930.28007
[11] Lu, N., Fractal Imaging (1997), Academic Press
[12] Mandelbrot, B., The fractal geometry of nature (1983), Freeman: Freeman San Francisco
[13] Nekka, F.; Li, J., Intersection of triadic Cantor sets with their translates. I. Fundamental properties, Chaos Solitons Fractals, 13, 9, 1807-1817 (2002) · Zbl 1002.28009
[14] Li, J.; Nekka, F., Intersection of triadic Cantor sets with their translates, II. Hausdorff measure spectrum function and its introduction for the classification of Cantor sets, Chaos Solitons Fractals, 19, 1, 35-46 (2004), Dedicated to our teacher, mentor and friend, Nobel laureate, Ilya Prigogine · Zbl 1068.28007
[15] Schief, A., Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122, 1, 111-115 (1994) · Zbl 0807.28005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.