Inverse measures, the inversion formula, and discontinuous multifractals (Q679034)

The authors introduce the notion of the inverse measure \(\mu^*\) to a given probability measure \(\mu\) on the unit interval \([0,1]\) in a language adopted for the physicist. If \(M\) is the distribution function of \(\mu\), requiring additionally \(M\) to be strictly increasing, then the inverse function \(M^*\) to \(M\) is also a distribution function of a certain measure \(\mu^*\) on \([0,1]\) and it is called the inverse measure. The interest of the authors is concentrated on the multifractal properties of both measures. For example, they sketch a heuristic argument for the inverse formula \[ f^*(\alpha)= \alpha f(1/\alpha), \] to be true where \(f\) and \(f^*\) are the associated multifractal spectra. Further comments are devoted, for example, to the inverse binomial measure, and finally, it will be discussed, what happens if the above additional property on \(M\) breaks down, i.e., \(\mu\) has zero measure intervals as the Cantor measure. Two forthcoming papers, making ideas rigorously, are announced.