Methods and Applications of Analysis

Volume 14 (2007)

Number 2

On Refinable Sets

Pages: 165 – 178



Xin-Rong Dai

Yang Wang


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

28A78, 28A80

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