Sub-self-similar sets

Author:
K. J. Falconer

Journal:
Trans. Amer. Math. Soc. **347** (1995), 3121-3129

MSC:
Primary 28A80; Secondary 28A78

MathSciNet review:
1264809

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Abstract | References | Similar Articles | Additional Information

Abstract: A compact set is called sub-self-similar if , where the 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.

**[1]**Christoph Bandt and Siegfried Graf,*Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure*, Proc. Amer. Math. Soc.**114**(1992), no. 4, 995–1001. MR**1100644**, 10.1090/S0002-9939-1992-1100644-3**[2]**Christoph Bandt,*Self-similar sets. I. Topological Markov chains and mixed self-similar sets*, Math. Nachr.**142**(1989), 107–123. MR**1017373**, 10.1002/mana.19891420107**[3]**Gerald A. Edgar,*Measure, topology, and fractal geometry*, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. MR**1065392****[4]**K. J. Falconer,*The geometry of fractal sets*, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR**867284****[5]**K. J. Falconer,*The Hausdorff dimension of self-affine fractals*, Math. Proc. Cambridge Philos. Soc.**103**(1988), no. 2, 339–350. MR**923687**, 10.1017/S0305004100064926**[6]**K. J. Falconer,*Dimensions and measures of quasi self-similar sets*, Proc. Amer. Math. Soc.**106**(1989), no. 2, 543–554. MR**969315**, 10.1090/S0002-9939-1989-0969315-8**[7]**-,*Fractal geometry--Mathematical foundations and applications*, Wiley, 1990.**[8]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[9]**Pertti Mattila,*Geometry of sets and measures in Euclidean spaces*, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR**1333890****[10]**R. Daniel Mauldin and S. C. Williams,*Hausdorff dimension in graph directed constructions*, Trans. Amer. Math. Soc.**309**(1988), no. 2, 811–829. MR**961615**, 10.1090/S0002-9947-1988-0961615-4**[11]**C. A. Rogers,*Hausdorff measures*, Cambridge University Press, London-New York, 1970. MR**0281862****[12]**Donald W. Spear,*Measures and self-similarity*, Adv. Math.**91**(1992), no. 2, 143–157. MR**1149621**, 10.1016/0001-8708(92)90014-C

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1264809-X

Article copyright:
© Copyright 1995
American Mathematical Society