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Transactions of the American Mathematical Society

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Sub-self-similar sets


Author: K. J. Falconer
Journal: Trans. Amer. Math. Soc. 347 (1995), 3121-3129
MSC: Primary 28A80; Secondary 28A78
DOI: https://doi.org/10.1090/S0002-9947-1995-1264809-X
MathSciNet review: 1264809
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Abstract: A compact set $ E \subseteq {{\mathbf{R}}^n}$ is called sub-self-similar if $ E \subseteq \bigcup\nolimits_{i = 1}^m {{S_i}(E)} $, where the $ {S_i}$ are similarity transfunctions. We consider various examples and constructions of such sets and obtain formulae for their Hausdorff and box dimensions, generalising those for self-similar sets.


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  • [1] C. Bandt and S. Graf, Self-similar sets $ 7$. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc. 114 (1992), 995-1001. MR 1100644 (93d:28014)
  • [2] C. Bandt, Self-similar sets I. Topological Markov chains and mixed self-similar sets, Math. Nachr. 142 (1989), 107-123. MR 1017373 (90j:54038)
  • [3] G. A. Edgar, Measure, topology and fractal geometry, Springer-Verlag, 1990. MR 1065392 (92a:54001)
  • [4] K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, 1985. MR 867284 (88d:28001)
  • [5] -, The Hausdorff dimension of self-affine fractals, Math. Proc. Cambridge Phil. Soc. 103 (1989), 339-350. MR 923687 (89h:28010)
  • [6] -, Dimensions and measures of quasi-self-similar sets, Proc. Amer. Math. Soc. 106 (1989), 543-554. MR 969315 (90c:58103)
  • [7] -, Fractal geometry--Mathematical foundations and applications, Wiley, 1990.
  • [8] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [9] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Univ. Press, 1995. MR 1333890 (96h:28006)
  • [10] R. D. Mauldin and S. C. Williams, Hausdorff dimensions in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), 811-829. MR 961615 (89i:28003)
  • [11] C. A. Rogers, Hausdorff measures, Cambridge Univ. Press, 1970. MR 0281862 (43:7576)
  • [12] D. W. Spear, Measure and self-similarity, Adv. Math. 91 (1992), 143-157. MR 1149621 (92k:28013)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1264809-X
Article copyright: © Copyright 1995 American Mathematical Society

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