Refinable distributions supported on self-affine tiles (Q1607514)

Let \(M\) be a linear transformation on \({\mathbb R}^n\) such that \(M{\mathbb Z}^n \subset {\mathbb Z}^n\) and all eigenvalues of \(M\) have the norms strictly larger than 1 \((M\) is called a dilation for \({\mathbb Z}^n)\). Let \(\Gamma\) be a coset representative of \({\mathbb Z}^n/M{\mathbb Z}^n\). The cardinality of \(\Gamma \) is \(|\text{det} M|.\) Let \(T\) be the unique compact set such that \[ MT = \bigcup_{k \in \Gamma} (T + k). \] Then \(T\) is called the self-affine tile determined by \(M\). A compactly supported distribution \(\phi\) on \({\mathbb R}^n\) is said to be refinable if it satisfies the refinement equation \[ \phi (\cdot) = \sum_{k \in {\mathbb Z}^n} c_k \phi(M\cdot - k)\;\;\text{ and } \;\widehat{\phi}(0) = 1, \] where \(\{c_k\}_{k \in {\mathbb Z}^n}\) has a finite length and satisfies \[ \sum_{k \in {\mathbb Z}^n} c_k = |\text{det} M|. \] Let \(\phi\) be a refinable distribution on \({\mathbb R}^n\) such that \(c_k = 0\) for \(k \notin \Gamma\). Then it is known that \(\phi\) is supported on the self-affine tile \(T\). In the paper under review, the author proves some conditions for \(\phi\) to be a Lebesgue-Stieltjes measure or to be absolutely continuous with respect to the Lebesgue measure.