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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniform perfectness of self-affine sets


Authors: Feng Xie, Yongcheng Yin and Yeshun Sun
Journal: Proc. Amer. Math. Soc. 131 (2003), 3053-3057
MSC (2000): Primary 28A78, 28A80
Published electronically: April 30, 2003
MathSciNet review: 1993212
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Abstract: Let $f_i(x)=A_ix+b_i (1\le i\le n)$ be affine maps of Euclidean space $\mathbb{R} ^N$ with each $A_i$ nonsingular and each $f_i$ contractive. We prove that the self-affine set $K$ of $\{f_1,\dots, f_n\}$ is uniformly perfect if it is not a singleton.


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Additional Information

Feng Xie
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China
Address at time of publication: 420 Temple St., #517, New Haven, Connecticut 06511
Email: xiefengmath@hotmail.com, feng.xie@yale.edu

Yongcheng Yin
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China – and – Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email: yin@math.zju.edu.cn

Yeshun Sun
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China
Email: sun@math.zju.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06976-4
PII: S 0002-9939(03)06976-4
Keywords: Uniformly perfect set, self-affine set, Hausdorff dimension
Received by editor(s): February 24, 2002
Published electronically: April 30, 2003
Additional Notes: This research was supported by the National Natural Science Foundation of China, Project No. 10171090.
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society