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Resistance Forms on Self-Similar Sets with Finite Ramification of Finite Type

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Abstract

In this paper, we introduce the finite neighboring type and the finite chain length conditions for a connected self-similar set K. We show that with these two conditions, K is a finitely ramified graph directed (f.r.g.d.) fractal defined by Hambly and Nyberg (Proc. Edinb. Math. Soc. (2) 46(1), 1–34 2003). We give some nontrivial examples and compute the harmonic structures on them explicitly. Furthermore, for a f.r.g.d. self-similar set K, we provide an equivalent description, the finitely ramified of finite type (f.r.f.t.) cell structure of K, and investigate the relationship of harmonic structures associated with different f.r.f.t. cell structures of K.

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Acknowledgments

We are grateful to the anonymous referee for the suggestions that led to the improvement of this paper. The research of Qiu was supported by the NSFC grant 11471157

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Authors and Affiliations

  1. Department of Mathematics, Cornell University, Ithaca, NY, 14853, USA

    Shiping Cao

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Hua Qiu

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  1. Shiping Cao
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  2. Hua Qiu
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Correspondence to Hua Qiu.

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Cao, S., Qiu, H. Resistance Forms on Self-Similar Sets with Finite Ramification of Finite Type. Potential Anal 54, 581–606 (2021). https://doi.org/10.1007/s11118-020-09840-w

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  • DOI: https://doi.org/10.1007/s11118-020-09840-w

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