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Dimension Functions of Self-Affine Scaling Sets
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- Journal:
- Canadian Mathematical Bulletin / Volume 56 / Issue 4 / 01 December 2013
- Published online by Cambridge University Press:
- 20 November 2018, pp. 745-758
- Print publication:
- 01 December 2013
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- Article
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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 
$n$ .
