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Relations among whitney sets, self-similar arcs and quasi-arcs

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Abstract

We study in this paper some relations among self-similar arcs, Whitney sets and quasi-arcs: we prove that any self-similar arc of dimension greater than 1 is a Whitney set; give a geometric sufficient condition for a self-similar arc to be a quasi-arc, and provide an example of a self-similar arc such that any subarc of it fails to be at-quasi-arc for anyt ≥ 1, which answers an open question on Whitney sets. We also show that self-similar arcs with the same Hausdorff dimension need not be Lipschitz equivalent.

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Authors and Affiliations

  1. Department of Mathematics, Tsinghua University, 100080, Beijing, P. R. China

    Zhi-Ying Wen

  2. Institute of Mathematics, Zhejiang Wanli University, 315100, Ningbo, P. R. China

    Li-Feng Xi

Authors
  1. Zhi-Ying Wen
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  2. Li-Feng Xi
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Correspondence to Zhi-Ying Wen.

Additional information

Supported by Special Funds for Major State Basic Research Projects of China, Morningside Center of Mathematics, NSFC (No. 10241003) and ZJNFS (No. 101026).

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Wen, ZY., Xi, LF. Relations among whitney sets, self-similar arcs and quasi-arcs. Isr. J. Math. 136, 251–267 (2003). https://doi.org/10.1007/BF02807200

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  • DOI: https://doi.org/10.1007/BF02807200

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