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Dimension Functions of Self-Affine Scaling Sets

Published online by Cambridge University Press:  20 November 2018

Xiaoye Fu  and
Jean-Pierre Gabardo
Xiaoye Fu
Affiliation:
Department of Mathematics, the Chinese University of Hong Kong, Hong Kong e-mail: xiaoyefu@gmail.com
Jean-Pierre Gabardo
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1 e-mail: gabardo@mcmaster.ca
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Abstract.

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In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\,\times \,n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK\,=\,\left( K\,+\,{{d}_{1}} \right)\,\cup \,\left( K\,+\,{{d}_{2}} \right)$, where $B\,=\,{{A}^{t}},\,A$ is an $n\,\times \,n$ integral expansive matrix with $\left| \det \,A \right|\,=\,2$, and ${{d}_{1}},\,{{d}_{2}}\,\in \,{{\mathbb{R}}^{n}}$. We show that the dimension function of $K$ must be constant if either $n\,=1$ or 2 or one of the digits is 0, and that it is bounded by $2\left| K \right|$ for any $n$.

Keywords

42C40scaling setself-affine tileorthonormal multiwaveletdimension function
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 745 - 758
Copyright
Copyright © Canadian Mathematical Society 2013

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