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Whitney's critical set in fractal

Yong Lin Information School, Renmin University of China, Beijing 100872, PR China Xi Lifeng Institute of Mathematics, Zhejiang Wanli University, Ningbo 315101, PR China

TBD mathscidoc:2207.43004

Chaos, Solitons & Fractals, 14, (7), 995-1006, 2002.10
The problem is concerned about how large (e.g. the Hausdorff dimension) is Whitney's critical set contained in a given fractal. For this, we prove that the Moran arc, an arc containing a Moran set, is a Whitney's critical set. The excellent open set condition is defined, when the condition holds, the associated self-similar set contains a Whitney's critical subset of full dimension. As its application, the Sierpinski gasket and Koch curve have Whitney's critical subset of full dimension. Finally, we provide a self-similar tree which never contains any Whitney's critical set.
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@inproceedings{yong2002whitney's,
  title={Whitney's critical set in fractal},
  author={Yong Lin, and Xi Lifeng},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707110404944580548},
  booktitle={Chaos, Solitons & Fractals},
  volume={14},
  number={7},
  pages={995-1006},
  year={2002},
}
Yong Lin, and Xi Lifeng. Whitney's critical set in fractal. 2002. Vol. 14. In Chaos, Solitons & Fractals. pp.995-1006. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707110404944580548.
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