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Connectedness of invariant sets of graph-directed IFS

  • Mathematics
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Wuhan University Journal of Natural Sciences

Abstract

In this paper, we study the connectedness of the invariant sets of a graph-directed iterated function system (IFS). For a graph-directed IFS with N states, we construct N graphs. We prove that all the invariant sets are connected, if and only if all the N graphs are connected; in this case, the invariant sets are all locally connected and path connected. Our result extends the results on the connectedness of the self-similar sets.

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References

  1. Hutchinson J E. Fractal and self similarity [J]. Indian Univ Math J, 1981, 30: 713–747.

    Article  Google Scholar 

  2. Falconer K. Fractal Geometry, Mathematical Foundations and Applications [M]. New York: John Wiley & Sons, 1990.

    Google Scholar 

  3. Hata M. On the structure of self-similar sets [J]. Japan J Appl Math, 1985, 2: 381–414.

    Article  Google Scholar 

  4. Kirat I, Lau K S. On the connectedness of self-affine tiles [J]. J Lond Math Soc, 2000, 62: 291–304.

    Article  Google Scholar 

  5. Bandt C, Rao H. Topology and separation of self-similar fractals in the plane [J]. Nonlinearity, 2007, 20: 1463–1474.

    Article  Google Scholar 

  6. Bandt C, Mesing M. Self-affine fractals of finite type [J]. Banach Center Publication, 2009, 84: 131–148.

    Article  Google Scholar 

  7. Lau K S, Luo J J, Rao H. Topological structure of fractal squares [J]. Math Proc Cambridge Philos Soc, 2013, 155: 73–86.

    Article  Google Scholar 

  8. Bedford T. Dimension and dynamics of fractal recurrent sets [J]. J London Math Soc, 1986, 33: 89–100.

    Article  Google Scholar 

  9. Mauldin D S, Williams S. Hausdorff dimension in graph directed constructions [J]. Trans Amer Math Soc, 1988, 309: 811–829.

    Article  Google Scholar 

  10. Falconer K. Techniques in Fractal Geometry [M]. Chichester: John Wiley & Sons, 1997.

    Google Scholar 

  11. Whyburn G T. Analytic Topology [M]. New York: American Mathematical Society Colloquium Publications, 1942:15–20.

    Book  Google Scholar 

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Authors and Affiliations

  1. Science and Technology College, Hubei University of Art and Science, Xiangyang 441025, Hubei, China

    Yanfang Zhang

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  1. Yanfang Zhang
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Correspondence to Yanfang Zhang.

Additional information

Foundation item: Supported by the Teaching Research Project of Hubei Province (2013469), and the 12th Five-Year Project of Education Plan of Hubei Province (2014B379)

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Zhang, Y. Connectedness of invariant sets of graph-directed IFS. Wuhan Univ. J. Nat. Sci. 21, 445–447 (2016). https://doi.org/10.1007/s11859-016-1194-1

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  • DOI: https://doi.org/10.1007/s11859-016-1194-1

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