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
Articles

Abstract

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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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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