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Volume 31, Issue 1
Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points

Shaoyuan Xu, Wangbin Xu & Zuoling Zhou

DOI: 10.4208/ata.2015.v31.n1.8

Anal. Theory Appl., 31 (2015), pp. 92-100.

Published online: 2017-01

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  • Abstract

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.

  • Keywords

Self-similar set, upper convex density, Hausdorff measure and Hausdorff dimension, contracting-similarity fixed point.

  • AMS Subject Headings

28A78, 28A80

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xushaoyuan@126.com ( Shaoyuan Xu)

  • BibTex
  • RIS
  • TXT
@Article{ATA-31-92, author = {Shaoyuan Xu , Wangbin Xu , and Zuoling Zhou , }, title = {Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {1}, pages = {92--100}, abstract = {

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n1.8}, url = {http://global-sci.org/intro/article_detail/ata/4625.html} }
TY - JOUR T1 - Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points AU - Shaoyuan Xu , AU - Wangbin Xu , AU - Zuoling Zhou , JO - Analysis in Theory and Applications VL - 1 SP - 92 EP - 100 PY - 2017 DA - 2017/01 SN - 31 DO - http://doi.org/10.4208/ata.2015.v31.n1.8 UR - https://global-sci.org/intro/article_detail/ata/4625.html KW - Self-similar set, upper convex density, Hausdorff measure and Hausdorff dimension, contracting-similarity fixed point. AB -

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.

Shaoyuan Xu , Wangbin Xu , and Zuoling Zhou , . (2017). Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points. Analysis in Theory and Applications. 31 (1). 92-100. doi:10.4208/ata.2015.v31.n1.8
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