Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Uniform perfectness of self-affine sets

Author(s): Feng Xie; Yongcheng Yin; Yeshun Sun
Journal: Proc. Amer. Math. Soc. 131 (2003), 3053-3057.
MSC (2000): Primary 28A78, 28A80
Posted: April 30, 2003
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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.

References:

[BP]
A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. 18 (1978), 475-483. MR 80a:30020

[CG]
L. Carleson and T. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 94h:30033

[Fa]
K. Falconer, Fractal geometry: Mathematical foundations and applications, Wiley, New York, 1990. MR 92j:28008

[HM]
A. Hinkkanen and G. J. Martin, Julia sets of rational semigroups, Math. Z. 222 (1996), no. 2, 161-169. MR 98d:30038

[JV]
P. Järvi and M. Vuorinen, Uniformly perfect sets and quasiregular mappings, J. London Math. Soc. 54 (1996), no. 2, 515-529. MR 98d:30031

[MC]
R. Mañé and L. F. da Rocha, Julia sets are uniformly perfect, Proc. Amer. Math. Soc. 116 (1992), 251-257. MR 92k:58229

[Ma]
P. Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge University Press, 1995. MR 96h:28006

[Po]
Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math 32 (1979), 192-199. MR 80j:30073

[S1]
R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2569-2575. MR 2000m:37075

[S2]
-, Uniformly perfect analytic and conformal attractor sets, Bull. London Math. Soc. 33 (2001), no. 3, 320-330. MR 2001m:37095

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 28A78, 28A80

Retrieve articles in all Journals with MSC (2000): 28A78, 28A80

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: 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
Posted: April 30, 2003
Additional Notes: This research was supported by the National Natural Science Foundation of China, Project No. 10171090.
Communicated by: Michael Handel
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google