Convolutions and the geometry of multifractal measures

Kenneth J Falconer and Toby C O'Neil

Appeared in Mathematische Nachrichten, 1999.


This paper relates multifractal features of a measure in \(\mathbb{R}^n\) to those of the projection of the measure onto typical \(m\)-dimensional linear subspaces. We achieve this through the introduction of suitably defined convolution kernels. This provides a unified approach to projections of measures and leads to new results on multifractal properties as well as alternative derivations of some existing formulae. These include formulae and estimates for the local dimensions and generalised \(q\)-dimensions of projected measures as well as more precise information about the limiting behaviour of multifractal expressions. We consider briefly how similar ideas may be applied to sections of a measure by \( (n-m)\)-dimensional planes.

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Created:21st May 1998
Modified:15 August 2000