Methods and Applications of Analysis

Volume 14 (2007)

Number 2

On Refinable Sets

Pages: 165 – 178

DOI: http://dx.doi.org/10.4310/MAA.2007.v14.n2.a3

Authors

Xin-Rong Dai

Yang Wang

Abstract

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.

Keywords

Hausdorff dimension; self-similar set; finite type condition

2010 Mathematics Subject Classification

28A78, 28A80

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