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Attractors of directed graph IFSs that are not standard IFS attractors and their Hausdorff measure

  • G. C. BOORE (a1) and K. J. FALCONER (a1)

For directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known.

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[1] E. Ayer and R. S. Strichartz Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc. 351 (1999), 37253741.

[4] R. Delaware Every set of finite Hausdorff measure is a countable union of sets whose Hausdorff measure and content coincide. Proc. Amer. Math. Soc. 131 (2002), 25372542.

[10] K. J. Falconer Fractal Geometry, Mathematical Foundations and Applications (John Wiley, Chichester, 2nd Ed.2003).

[11] De-J. Feng and Y. Wang On the structures of generating iterated function systems of Cantor sets. Adv. Math. 222 (2009), 19641981.

[12] J. Hutchinson Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713747.

[14] R. D. Mauldin and S. C. Williams Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309 (1988), 811829.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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