On the boundary of attractors with non-void interior期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the boundary of attractors with non-void interior

Authors:	Ka-Sing Lau You Xu

Institution:	Department of Mathematics, The Chinese University of Hong Kong, Hong Kong ; Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Abstract:	Let $\left\{ f_i\right\} _{i=1}^N$ be a family of contractive mappings on $\mathbb{R}^{d\text{ }}$ such that the attractor has nonvoid interior. We show that if the 's are injective, have non-vanishing Jacobian on , 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 has measure zero. In addition if the '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.

Keywords:	Boundary Hausdorff dimension self-affine tiles self-similarity singular values

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