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

 

On the boundary of attractors
with non-void interior


Authors: Ka-Sing Lau and You Xu
Journal: Proc. Amer. Math. Soc. 128 (2000), 1761-1768
MSC (2000): Primary 28A80, 52C22; Secondary 28A78
Published electronically: October 27, 1999
MathSciNet review: 1662265
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\left\{ f_i\right\} _{i=1}^N$ be a family of $N$ contractive mappings on $\mathbb{R}^{d\text{ }}$ such that the attractor $K$ has nonvoid interior. We show that if the $f_i$'s are injective, have non-vanishing Jacobian on $K$, and $f_i\left( K\right) \cap f_j\left( K\right) $ have zero Lebesgue measure for $i\neq j,$ then the boundary $\partial K$ of $K$ has measure zero. In addition if the $f_i$'s are affine maps, then the conclusion can be strengthened to $\dim _H\left( \partial K\right) <d$. These improve a result of Lagarias and Wang on self-affine tiles.


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Additional Information

Ka-Sing Lau
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Email: kslau@math.cuhk.edu.hk

You Xu
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: yoxst+@pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05303-4
PII: S 0002-9939(99)05303-4
Keywords: Boundary, Hausdorff dimension, self-affine tiles, self-similarity, singular values
Received by editor(s): January 8, 1998
Received by editor(s) in revised form: July 23, 1998
Published electronically: October 27, 1999
Additional Notes: The first author was partially supported by the RGC grant CUHK4057/98P
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society