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Discrete & Computational Geometry

, Volume 46, Issue 2, pp 389–393 | Cite as

On Lebesgue Measure of Integral Self-Affine Sets

  • Ievgen V. Bondarenko
  • Rostyslav V. Kravchenko
Article

Abstract

Let A be an expanding integer n×n matrix and D be a finite subset of ℤ n . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈ℤ n , and the measure of the intersection of self-affine sets T(A,D 1)∩T(A,D 2) for different sets D 1, D 2⊂ℤ n .

Keywords

Self-affine set Tile Graph-directed system Self-similar action 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Ievgen V. Bondarenko
    • 1
  • Rostyslav V. Kravchenko
    • 2
  1. 1.National Taras Shevchenko University of KievKievUkraine
  2. 2.Texas A&M UniversityCollege StationUSA

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