On the structures of generating iterated function systems of Cantor sets. (English) Zbl 1184.28013
Let \(\Phi=\{\phi_j\}_{1\leq j\leq N}\) be an IFS in \(\mathbb R^d\), and \(F_\Phi\subset\mathbb R^d\) the attractor of \(\Phi\).
The following question is rather fundamental and difficult, and is open for a long time in fractal geometry: is it possible to express \(F_\Phi\) as the attractor of another IFS? The close relative question is: if \(\Psi=\{\psi_j\}_{1\leq j\leq M}\) be another IFS such that \(F_\Phi=F_\Psi\), what are the relations between these two IFS.
For studying the problem above, the authors introduce the notions of generating IFSs, minimal generating IFS, logarithmic commensurability, They study in details the semi-group of homogeneous generating IFSs of a Cantor set in \(\mathbb R\) under the open set condition. They prove that if \(\dim_HF<1\), then all generating IFSs must have logarithmically commensurable contraction factors. From this result, the authors derive a structure theorem for the semi-group of generating IFSs under open set condition. These results are essentially the first step of studies in this direction. The authors give also some non-trivial examples in the paper.
The following question is rather fundamental and difficult, and is open for a long time in fractal geometry: is it possible to express \(F_\Phi\) as the attractor of another IFS? The close relative question is: if \(\Psi=\{\psi_j\}_{1\leq j\leq M}\) be another IFS such that \(F_\Phi=F_\Psi\), what are the relations between these two IFS.
For studying the problem above, the authors introduce the notions of generating IFSs, minimal generating IFS, logarithmic commensurability, They study in details the semi-group of homogeneous generating IFSs of a Cantor set in \(\mathbb R\) under the open set condition. They prove that if \(\dim_HF<1\), then all generating IFSs must have logarithmically commensurable contraction factors. From this result, the authors derive a structure theorem for the semi-group of generating IFSs under open set condition. These results are essentially the first step of studies in this direction. The authors give also some non-trivial examples in the paper.
Reviewer: Hua Su (Beijing)
Keywords:
generating IFS; minimal IFS; convex open sets condition; logarithmic commensurability; Hausdorff dimension; self-similar setsReferences:
| [1] | Ayer, E.; Strichartz, R. S., Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Amer. Math. Soc., 351, 3725-3741 (1999) · Zbl 0933.28003 |
| [2] | Bandt, C.; Retta, T., Topological spaces admitting a unique fractal structure, Fund. Math., 141, 257-268 (1992) · Zbl 0832.28011 |
| [3] | Barnsley, M. F.; Hurd, L. P., Fractal Image Compression (1993), A.K. Peters, Ltd.: A.K. Peters, Ltd. Wellesley, MA, illustrations by Louisa F. Anson · Zbl 0796.68186 |
| [4] | Croft, H. T.; Falconer, K. J.; Guy, R. K., Unsolved Problems in Geometry (1994), Springer-Verlag: Springer-Verlag New York |
| [5] | Deliu, A.; Geronimo, J.; Shonkwiler, R., On the inverse fractal problem for two-dimensional attractors, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 355, 1017-1062 (1997) · Zbl 0886.58068 |
| [6] | Elekes, M.; Keleti, T.; Máthé, A., Self-similar and self-affine sets; measure of the intersection of two copies, preprint · Zbl 1200.28005 |
| [7] | Falconer, K. J., The Geometry of Fractal Sets (1985), Cambridge University Press · Zbl 0587.28004 |
| [8] | Falconer, K. J.; O’Connor, J. J., Symmetry and enumeration of self-similar fractals, Bull. Lond. Math. Soc., 39, 272-282 (2007) · Zbl 1148.28006 |
| [9] | D.J. Feng, H. Rao, Y. Wang, Self-similar subsets of the Cantor set, in preparation; D.J. Feng, H. Rao, Y. Wang, Self-similar subsets of the Cantor set, in preparation · Zbl 1321.28010 |
| [10] | Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J., 30, 713-747 (1981) · Zbl 0598.28011 |
| [11] | Lagarias, J. C.; Wang, Y., Integral self-affine tiles in \(R^n\). I. Standard and nonstandard digit sets, J. London Math. Soc. (2), 54, 161-179 (1996) · Zbl 0893.52014 |
| [12] | Lu, N., Fractal Imaging (1997), Academic Press · Zbl 0981.68167 |
| [13] | Csörnyei, M., Open questions, Period. Math. Hungar., 37, 227-237 (1998) |
| [14] | Marion, J., Mesure de Hausdorff d’un fractal à similitude interne, Ann. Sci. Math. Québec, 10, 51-84 (1986) · Zbl 0613.28007 |
| [15] | Odlyzko, A. M., Nonnegative digit sets in positional number systems, Proc. London Math. Soc. (3), 37, 213-229 (1978) · Zbl 0391.10012 |
| [16] | Peres, Y.; Shmerkin, P., Resonance between Cantor sets, Ergodic Theory Dynam. Systems, 29, 201-221 (2009) · Zbl 1159.37005 |
| [17] | P. Shmerkin, personal communication; P. Shmerkin, personal communication |
| [18] | Strichartz, R., Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355, 4019-4043 (2003) · Zbl 1041.28006 |
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.
