Separation properties for self-similar sets
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- by Andreas Schief PDF
- Proc. Amer. Math. Soc. 122 (1994), 111-115 Request permission
Abstract:
Given a self-similar set K in ${\mathbb {R}^s}$ we prove that the strong open set condition and the open set condition are both equivalent to ${H^\alpha }(K) > 0$, where $\alpha$ is the similarity dimension of K and ${H^\alpha }$ denotes the Hausdorff measure of this dimension. As an application we show for the case $\alpha = s$ that K possesses inner points iff it is not a Lebesgue null set.References
- 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, DOI 10.1090/S0002-9939-1992-1100644-3
- Claude Berge, Graphs and hypergraphs, North-Holland Mathematical Library, Vol. 6, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Translated from the French by Edward Minieka. MR 0357172
- K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677 S. Graf, The equidistribution on self-similar sets, MIP-8929 Passau, 1989.
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15–23. MR 14397, DOI 10.1017/s0305004100022684
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 111-115
- MSC: Primary 28A80; Secondary 28A78
- DOI: https://doi.org/10.1090/S0002-9939-1994-1191872-1
- MathSciNet review: 1191872