Inversion formula for continuous multifractals
In their previous paper [Adv. Appl. Math. 18, No. 1, 50-58 (1977; Zbl 0870.28004)] the authors introduced the inverse measure of a multifractal \(\mu\) and argued heuristically that their multifractal spectra are linked by the ``inversion formula''. In this paper, they establish rigorously the inversion formula for the Hausdorff, packing and coarse grained spectra under the assumption that both \(\mu\) and its inverse measure are continuous. In the discontinuous case, the authors consider especially discontinuous self-similar measures and show that the inversion formula still holds for the Hausdorff and packing spectra, but it does not hold for the coarse grained spectrum. The details of these and further related results are presented in their paper in Math. Proc. Camb. Philos. Soc. 123, No. 1, 133-157 (1998; following review).
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