Monatshefte für Mathematik

, Volume 169, Issue 3–4, pp 397–407 | Cite as

Spectral self-affine measures with prime determinant



The self-affine measure \(\mu _{M,D}\) relating to an expanding matrix \(M\in M_{n}(\mathbb Z )\) and a finite digit set \(D\subset \mathbb Z ^n\) is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of \(\mu _{M,D}\) in the case when \(|\det (M)|=p\) is a prime. The main result shows that under certain mild conditions, if there are two points \(s_{1}, s_{2}\in \mathbb R ^{n}, s_{1}-s_{2}\in \mathbb Z ^{n}\) such that the exponential functions \(e_{s_{1}}(x), e_{s_{2}}(x)\) are orthogonal in \(L^{2}(\mu _{M,D})\), then the self-affine measure \(\mu _{M,D}\) is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.


Iterated function system Self-affine measure Spectrality 

Mathematics Subject Classification (2010)

28A80 42C05 46C05 


  1. 1.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Fuglede, B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101–121 (1974)Google Scholar
  3. 3.
    Li, J.-L.: Spectral sets and spectral self-affine measures, Ph.D. Thesis, The Chinese University of Hong Kong, November, 2004Google Scholar
  4. 4.
    Dutkay, D.E., Jorgensen, P.E.T.: Fourier series on fractals: a parallel with wavelet theory, in: Radon transform, geometry, and wavelets. Contemp. Math. 464, 75–101 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dutkay, D.E., Jorgensen, P.E.T.: Duality questions for operators, spectrum and measures. Acta Appl. Math. 108, 515–528 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Jorgensen, P.E.T., Pedersen, S.: Dense analytic subspaces in fractal \(L^2-\)spaces. J. Anal. Math. 75, 185–228 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Strichartz, R.: Convergence of mock Fourier series. J. Anal. Math. 99, 333–353 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dutkay, D.E., Jorgensen, P.E.T.: Fourier frequencies in affine iterated function systems. J. Funct. Anal. 247, 110–137 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dutkay, D.E., Jorgensen, P.E.T.: Probability and Fourier duality for affine iterated function systems. Acta Appl. Math. 107, 293–311 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Łaba, I., Wang, Y.: On spectral Cantor measures. J. Funct. Anal. 193, 409–420 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Li, J.-L.: Spectral self-affine measures in \(\mathbb{R}^{n}\). Proc. Edinburgh Math Soc. 50, 197–215 (2007)CrossRefGoogle Scholar
  12. 12.
    Li, J.-L.: \(\mu _{M, D}-\)Orthogonality and compatible pair. J. Funct. Anal. 244, 628–638 (2007)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Dutkay, D.E., Jorgensen, P.E.T.: Iterated function systems, Ruelle operators, and invariant projetive measures. Math. Comp. 75, 1931–1970 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Li, J.-L., Wen, Z.-Y.: Spectrality of planar self-affine measures with two-element digit set. Sci. China Math. 55, 593–605 (2012)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Li, J.-L.: Singularity of certain self-affine measures. J. Math. Anal. Appl. 347, 375–380 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anPeople’s Republic of China

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