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# Dimension Functions of Self-Affine Scaling Sets

Published:2013-01-10
Printed: Dec 2013
• Xiaoye Fu,
Department of Mathematics, the Chinese University of Hong Kong, Hong Kong
• Jean-Pierre Gabardo,
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S4K1
 Format: LaTeX MathJax PDF

## Abstract

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 $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, 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 $2\lvert K\rvert$ for any $n$.
 Keywords: scaling set, self-affine tile, orthonormal multiwavelet, dimension function
 MSC Classifications: 42C40 - Wavelets and other special systems