On the boundary of attractors with non-void interior
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- by Ka-Sing Lau and You Xu PDF
- Proc. Amer. Math. Soc. 128 (2000), 1761-1768 Request permission
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.References
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Additional Information
- Ka-Sing Lau
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
- MR Author ID: 190087
- Email: kslau@math.cuhk.edu.hk
- You Xu
- Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- Email: yoxst+@pitt.edu
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1761-1768
- MSC (2000): Primary 28A80, 52C22; Secondary 28A78
- DOI: https://doi.org/10.1090/S0002-9939-99-05303-4
- MathSciNet review: 1662265