Advertisement

Monatshefte für Mathematik

, Volume 162, Issue 3, pp 355–374 | Cite as

The quantization dimension of the self-affine measures on general Sierpiński carpets

  • Sanguo Zhu
Article

Abstract

Let μ be a self-affine measure on a general Sierpiński carpet E. We give a characterization for the upper and lower quantization dimension of μ in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D r (μ) of μ exists for all r > 0 under an additional condition. We establish an explicit formula for D r (μ) and show that it increases to the box-counting dimension \({dim_B^* \mu}\) of μ as r tends to infinity. For a class of Sierpiński carpets E and the uniform measures μ on E, we show that the quantization dimension of μ coincides with its box-counting dimension and that the D r (μ)-dimensional upper and lower quantization coefficient of μ are both positive and finite.

Keywords

Quantization dimension Quantization coefficient Self-affine measure Sierpiński carpets Finite maximal anti-chain 

Mathematics Subject Classification (2000)

Primary 28A80 28A78 Secondary 94A15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bedford, T.: Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD Thesis, University of Warwick (1984)Google Scholar
  2. 2.
    Bucklew J.A., Wise G.L.: Multidimensional asymptotic quantization with rth power distortion measures. IEEE Trans. Inform. Theory 28, 239–247 (1982)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Falconer K.J.: Generalized dimensions of measures on self-affine sets. Nonlinearity 12, 877–891 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Graf S.: On Bandt’s tangential distribution for self-similar measures. Monatsh. Math. 120, 223–246 (1995)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Graf, S., Luschgy, H.: Foundations of quantization for probability distributons. Lecture Notes in Mathematis, vol. 1730. Springer, Berlin (2000)Google Scholar
  6. 6.
    Graf S., Luschgy H.: The quantization dimension of self-similar probabilities. Math. Nachr. 241, 103–109 (2002)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Graf S., Luschgy H.: The point density measure in the quantization of self-similar probabilities. Math. Proc. Camb. Phil. Soc. 136, 687–717 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gray R., Neuhoff D.: Quantization. IEEE Trans. Inform. Theory 44, 2325–2383 (1998)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hua S.: On the dimension of generalized self-similar sets. Acta Math. Appl. Sin. 17(4), 551–558 (1994)Google Scholar
  10. 10.
    Hua S., Li W.X.: Packing dimension of generalized Moran sets. Progr. Nat. Sci. 6(2), 148–152 (1996)MathSciNetGoogle Scholar
  11. 11.
    Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    King J.F.: The singularity spectrum for general Sierpński carpets. Adv. Math. 116, 1–11 (1995)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Lindsay L.J., Mauldin R.D.: Quantization dimension for conformal systems. Nonlinearity 15, 189–199 (2002)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Mcmullen C.: The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96, 1–9 (1984)MathSciNetMATHGoogle Scholar
  15. 15.
    Peres Y.: The packing measure of self-affine carpets. Math. Proc. Camb. Phil. Soc. 115, 437–450 (1994)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Peres Y.: The self-affine carpets of Mcmullen and Bedford have infinite Hausdorff measure. Math. Proc. Camb. Phil. Soc. 116, 513–526 (1994)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Pötzelberger K.: The quantization dimension of distributions. Math. Proc. Camb. Phil. Soc. 131, 507–519 (2001)MATHGoogle Scholar
  18. 18.
    Zador, P.L.: Development and evaluation of procedures for quantizing multivariate distributions. PhD Thesis, Stanford University (1964)Google Scholar
  19. 19.
    Zhu S.: Quantization dimension of probability measures supported on Cantor-like sets. J. Math. Anal. Appl. 338, 742–750 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations