Mathematische Zeitschrift

, Volume 283, Issue 1–2, pp 39–58 | Cite as

On the quantization for self-affine measures on Bedford–McMullen carpets

Article

Abstract

For a self-affine measure on a Bedford–McMullen carpet we prove that its quantization dimension of order \(r>0\) exists and determine its exact value. Further, we give various sufficient conditions for the corresponding upper and lower quantization coefficient to be both positive and finite. Finally, we compare the quantization dimension with corresponding quantities derived from the multifractal free energy function and show that—different from conformal systems—they in general do not coincide.

Keywords

Quantization dimension Quantization coefficient Bedford–McMullen carpets Self-affine measures Multifractal formalism 

Mathematics Subject Classification

28A75 28A80 94A15 

References

  1. 1.
    Bedford, T.: Crinkly curves, Markov partitions and box dimensions in self-similar sets. Ph.D. thesis, University of Warwick (1984)Google Scholar
  2. 2.
    Bucklew, J.A., Wise, G.L.: Multidimensional asymptotic quantization with \(r\)th power distortion measures. IEEE Trans. Inform. Theory 28, 239–247 (1982)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Falconer, K.J.: Techniques in Fractal Geometry. Wiley, London (1997)MATHGoogle Scholar
  4. 4.
    Falconer, K.J.: Generalized dimensions of measures on almost self-affine sets. Nonlinearity 23, 1047–1069 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Graf, S.: On Bandt’s tangential distribution for self-similar measures. Monatsh. Math. 120, 223–246 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Graf, S., Luschgy, H.: Foundations of quantization for probability distributions. Lecture Notes in Mathematics, vol. 1730. Springer (2000)Google Scholar
  7. 7.
    Graf, S., Luschgy, H.: Asymptotics of the quantization error for self-similar probabilities. Real. Anal. Exch. 26, 795–810 (2001)MathSciNetMATHGoogle Scholar
  8. 8.
    Graf, S., Luschgy, H.: Quantization for probabilitiy measures with respect to the geometric mean error. Math. Proc. Camb. Philos. Soc. 136, 687–717 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Graf, S., Luschgy, H.: The point density measure in the quantization of self-similar probabilities. Math. Proc. Camb. Philos. Soc. 138, 513–531 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gray, R., Neuhoff, D.: Quantization. IEEE Trans. Inform. Theory 44, 2325–2383 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gui, Y., Li, W.X.: Multiscale self-affine Sierpinski carpets. Nonlinearity 23, 495–512 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jordan, T., Rams, M.: Multifractal analysis for Bedford–McMullen carpets. Math. Proc. Camb. Philos. Soc. 150, 147–156 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kesseböhmer, M., Zhu, S.: Some recent developments in quantization of fractal measures. In: Bandt, C., Falconer, K., Zähle, M. (eds.) Fractal Geometry and Stochastics V, Progress in Probability, vol. 70, pp. 105–120 (2015). Springer, Switzerland (2015). doi:10.1007/978-3-319-18660-3_7
  15. 15.
    King, J.F.: The singularity spectrum for general Sierpiński carpets. Adv. Math. 116, 1–11 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kreitmeier, W.: Optimal quantization for dyadic homogeneous Cantor distributions. Math. Nachr. 281, 1307–1327 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lalley, S.P., Gatzouras, D.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41, 533–568 (1992)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lindsay, L.J., Mauldin, R.D.: Quantization dimension for conformal iterated function systems. Nonlinearity 15, 189–199 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    McMullen, C.: The Hausdorff dimension of general Sierpiński carpetes. Nagoya Math. J. 96, 1–9 (1984)MathSciNetMATHGoogle Scholar
  20. 20.
    Peres, Y.: The packing measure of self-affine carpets. Math. Proc. Camb. Philos. Soc. 115, 437–450 (1994)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Peres, Y.: The self-affine carpetes of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Camb. Philos. Soc. 116, 513–526 (1994)CrossRefMATHGoogle Scholar
  22. 22.
    Pötzelberger, K.: The quantization dimension of distributions. Math. Proc. Camb. Philos. Soc. 131, 507–519 (2001)MathSciNetMATHGoogle Scholar
  23. 23.
    Zador, P.L.: Development and evaluation of procedures for quantizing multivariate distributions. Ph.D. thesis, Stanford University (1964)Google Scholar
  24. 24.
    Zhu, S.: Quantization dimension of probability measures supported on Cantor-like sets. J. Math. Anal. Appl. 338, 742–750 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Zhu, S.: The quantization dimension for self-affine measures on general Sierpiński carpets. Monatsh. Math. 162, 355–374 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Fachbereich 3 – Mathematik und InformatikUniversität BremenBremenGermany
  2. 2.School of Mathematics and PhysicsJiangsu University of TechnologyChangzhouChina

Personalised recommendations