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Open set condition for graph directed self-similar structure

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Abstract

Graph directed self-similar structure generalizes the concept of self-similar set and contains some important instants of fractal sets. We characterize the open set condition (OSC), which is fundamental in the study of self-similar set, for graph directed self-similar structure in terms of the post critical set. Using this characterization, we establish the relations between OSC and other separation conditions including post-critically finite, finitely ramified condition and finite preimage property. It turns out that whether the intrinsic metric is doubling makes difference. In particular, finitely ramified condition implies OSC in case of doubling metric but does not in case of non-doubling metric.

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Acknowledgments

The work is supported partially by the NNSF No. 11271223 and No. 61071066. The first author is supported by Renmin University New Teacher Fund No. 10XNF038.

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Authors and Affiliations

  1. School of Information, Renmin University, Beijing, 100872, People’s Republic of China

    Tian-jia Ni

  2. Department of Mathematical Science, Tsinghua University, Beijing, 100084, People’s Republic of China

    Zhi-ying Wen

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  1. Tian-jia Ni
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  2. Zhi-ying Wen
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Correspondence to Tian-jia Ni.

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Ni, Tj., Wen, Zy. Open set condition for graph directed self-similar structure. Math. Z. 276, 243–260 (2014). https://doi.org/10.1007/s00209-013-1196-z

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  • DOI: https://doi.org/10.1007/s00209-013-1196-z

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