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Mathematische Annalen

, Volume 319, Issue 4, pp 817–836 | Cite as

Scaling properties of Hausdorff and packing measures

  • Marianna Csörnyei
  • R. Daniel Mauldin
Original article

Abstract.

Let \(m\in{\mathbb N}\). Let \(\theta\) be a continuous increasing function defined on \({\mathbb R}^+\), for which \(\theta(0)=0\) and \(\theta(t)/t^m\) is a decreasing function of t. Let \(\|\cdot\|\) be a norm on \({\mathbb R}^m\), and let \(\rho\), \({\mathcal H}^\theta ={\mathcal H}^\theta_\rho\), \({\mathcal P}^\theta ={\mathcal P}^\theta_\rho\) denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions \(\theta\) such that the corresponding Hausdorff or packing measure scales with exponent \(\alpha\) by showing it must be of the form \(\theta(t) = t^\alpha L(t)\), where L is slowly varying. We also show that for continuous increasing functions \(\theta\) and \(\eta\) defined on \({\mathbb R}^+\), for which \(\theta(0)=\eta(0)=0\), \({\mathcal H}^\theta={\mathcal P}^\eta\) is either trivially true or false: we show that if \({\mathcal H}^\theta={\mathcal P}^\eta\), then \({\mathcal H}^\theta={\mathcal P}^\eta=c\cdot\lambda\) for a constant c, where \(\lambda\) is the Lebesgue measure on \({\mathbb R}^m\).

Mathematics Subject Classification (2000): 28A12, 28A80 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Marianna Csörnyei
    • 1
  • R. Daniel Mauldin
    • 2
  1. 1.Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK (e-mail: mari@math.ucl.ac.uk) GB
  2. 2.Department of Mathematics, University of North Texas, Box 311430, Denton, TX 76203, USA (e-mail: mauldin@unt.edu) US

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