Dimension Functions of Self-Affine Scaling Sets

Fu, Xiaoye ; Gabardo, Jean-Pierre

Canadian mathematical bulletin, Tome 56 (2013) no. 4, pp. 745-758

Résumé

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 $n$ .

Détail

DOI : 10.4153/CMB-2012-040-4

Mots-clés : 42C40, scaling set, self-affine tile, orthonormal multiwavelet, dimension function