×

zbMATH — the first resource for mathematics

Separation properties for self-similar sets. (English) Zbl 0807.28005
Summary: 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.

MSC:
28A78 Hausdorff and packing measures
28A80 Fractals
PDF BibTeX XML Cite
Full Text: DOI
References:
[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. · Zbl 0823.28003
[2] Claude Berge, Graphs and hypergraphs, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Translated from the French by Edward Minieka; North-Holland Mathematical Library, Vol. 6. · Zbl 0254.05101
[3] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. · Zbl 0587.28004
[4] Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. · Zbl 0689.28003
[5] S. Graf, The equidistribution on self-similar sets, MIP-8929 Passau, 1989.
[6] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713 – 747. · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055 · doi.org
[7] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15 – 23. · Zbl 0063.04088
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.