Skip to main content
SpringerLink
Log in

Quantization dimension and temperature function for bi-Lipschitz mappings

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The quantization dimension function for a probability measure supported on the limit set generated by a set of contractive bi-Lipschitz mappings is determined. The relationship between the quantization dimension function and the temperature function of the thermodynamic formalism arising in multifractal analysis is established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. J. A. Bucklew and G. L. Wise, Multidimensional asymptotic quantization with rth power distortion measures, Institute of Electrical and Electronics Engineers. Transactions on Information Theory 28 (1982), 239–247.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. J. Falconer, Techniques in Fractal Geometry, Wiley, Chichester, 1997.

    MATH  Google Scholar 

  3. K. J. Falconer, The multifractal spectrum of statistically self-similar measures, Journal of Theoretical Probability 7 (1994), 681–701.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression, Kluwer Academic, Boston, 1992.

    Book  MATH  Google Scholar 

  5. R. Gray and D. Neuhoff, Quantization, Institute of Electrical and Electronics Engineers. Transactions on Information Theory 44 (1998), 2325–2383.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics 1730, Springer, Berlin, 2000.

    Google Scholar 

  7. S. Graf and H. Luschgy, The quantization dimension of self-similar probabilities, Mathematische Nachrichten 241 (2002), 103–109.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Hutchinson, Fractals and self-similarity, Indiana University Mathematics Journal 30 (1981), 713–747.

    Article  MathSciNet  MATH  Google Scholar 

  9. T. Halsey, M. Jensen, L. Kadanoff and I. Procaccia, Fractal measures and their singularities: the characterization of strange sets, Physical Review. A 33 (1986), 1141–1151.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. J. Lindsay and R. D. Mauldin, Quantization dimension for conformal iterated function systems, Nonlinearity 15 (2002), 189–199.

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Patzschke, Self-conformal multifractal measures, Advances in Applied Mathematics 19 (1997), 486–513.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. K. Roychowdhury, Quantization dimension function and ergodic measure with bounded distortion, Bulletin of the Polish Academy of Sciences. Mathematics 57 (2009), 251–262.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. K. Roychowdhury, Quantization dimension for some Moran measures, Proceedings of the American Mathematical Society 138 (2010), 4045–4057.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. K. Roychowdhury, Quantization dimension function and Gibbs measure associated with Moran set, Journal of Mathematical Analysis and Applications 373 (2011), 73–82.

    Article  MathSciNet  MATH  Google Scholar 

  15. X. Wang and M. Dai, Mixed quantization dimension function and temperature function for conformal measures, International Journal of Nonlinear Science 10 (2010), 24–31.

    MathSciNet  MATH  Google Scholar 

  16. P. L. Zador, Development and evaluation of procedures for quantizing multivariate distributions, Ph.D. Thesis, Stanford University, 1964.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Texas-Pan American, 1201 West University Drive, Edinburg, TX, 78539-2999, USA

    Mrinal Kanti Roychowdhury

Authors
  1. Mrinal Kanti Roychowdhury
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mrinal Kanti Roychowdhury.

Additional information

Dedicated to the memory of Professor Daniel J. Rudolph

Rights and permissions

Reprints and permissions

About this article

Cite this article

Roychowdhury, M.K. Quantization dimension and temperature function for bi-Lipschitz mappings. Isr. J. Math. 192, 473–488 (2012). https://doi.org/10.1007/s11856-012-0035-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0035-y

Keywords

Search

Navigation