Methods and Applications of Analysis
Volume 14 (2007)
On Refinable Sets
Pages: 165 – 178
A refinable set is a compact set with positive Lebesgue measure whose characteristic function satisfies a refinement equation. Refinable sets are a generalization of self-affine tiles. But unlike the latter, the refinement equations defining refinable sets may have negative coefficients, and a refinable set may not tile. In this paper, we establish some fundamental properties of these sets.
Hausdorff dimension; self-similar set; finite type condition
2010 Mathematics Subject Classification