Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure

Authors: Christoph Bandt and Siegfried Graf
Journal: Proc. Amer. Math. Soc. 114 (1992), 995-1001
MSC: Primary 28A80; Secondary 58F08
MathSciNet review: 1100644
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For self-similar sets with nonoverlapping pieces, Hausdorff dimension and measure are easily determined. We express "absence of overlap" in terms of discontinuous action of a family of similitudes, thus improving the usual "open set condition".

References [Enhancements On Off] (What's this?)

  • [1] C. Bandt, Self-similar sets 5. Integer matrices and tilings of $ {\mathbb{R}^n}$, Proc. Amer. Math. Soc. 112 (1991), 549-562. MR 1036982 (92d:58093)
  • [2] C. Berge, Graphs and hypergraphs, North-Holland, Amsterdam, 1973. MR 0357172 (50:9640)
  • [3] G. A. Edgar, Measure, topology, and fractal geometry, Springer-Verlag, New York, 1990. MR 1065392 (92a:54001)
  • [4] K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, 1985; Fractal geometry, Wiley, New York, 1990. MR 867284 (88d:28001)
  • [5] M. Hata, On the structure of self-similar sets, Japan J. Appl. Math. 2 (1985), 381-414. MR 839336 (87g:58080)
  • [6] J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
  • [7] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15-23. MR 0014397 (7:278f)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A80, 58F08

Retrieve articles in all journals with MSC: 28A80, 58F08

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society