Journal of Statistical Physics

, Volume 86, Issue 3–4, pp 707–720 | Cite as

On equivalent definitions of the correlation dimension for a probability measure

  • Charles-Antoine Guerin
  • Matthias Holschneider


In mathematical physics, one sometimes has to deal with averages of the type
$$M\mu (T) = \frac{1}{{T^n }}\int\limits_{|\xi | \leqslant T} { d\xi |\hat \mu (\xi )|^2 , T > 0}$$
where\(\hat \mu\) is the Fourier transform of some probability Borel measure μ. We show that the asymptotic behavior ofMμ is governed by the usual (upper and lower) correlation dimension of the measure μ.

Key Words

Fractal measure correlation dimension 


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  1. 1.
    C. Allain and M. Cloitre, Optical diffraction on fractals,Phys. Rev. B 33(5):3566–3569 (1985).Google Scholar
  2. 2.
    C. A. Guerin and M. Holschneider, Scattering on fractal measures,J. Phys. A: Math. Gen. 29:1–17 (1996).Google Scholar
  3. 3.
    M. Holschneider, Fractal wavelet dimension and localisation,Commun. Math. Phys. 160:457–473 (1994).Google Scholar
  4. 4.
    R. Ketzmerick, G. Petschel, and T. Geisel, Slow decay of temporal correlations in quantum systems with Cantor spectral,Phys. Rev. Lett. 69(5):695–698 (1992).Google Scholar
  5. 5.
    Ya. B. Pesin, On rigorous mathematical definition of the correlation dimension and generalized spectrum for dimension,J. Stat. Phys. 71(3/4):529–547 (1993).Google Scholar
  6. 6.
    M. Reed and B. Simon,Methods of Modern Mathematical Physics: Functional Analysis (Academic Press, New York, 1980).Google Scholar
  7. 7.
    W. Rudin,Analyse réelle et complexe (Masson, Paris, 1978).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Charles-Antoine Guerin
    • 1
  • Matthias Holschneider
    • 1
  1. 1.Centre de Physique ThéoriqueCNRS-LuminyMarseille Cedex 9France

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