Non-conservative character of the intersection of self-similar cascades

When a self-similar cascade is intersected, the resulting cascade process generating the intersection set is in general non-conservative, i.e. in the fragmentation process the related measure is not conserved. It is shown that the non-conservative character of a cascade invalidates the experimental analysis of the process. In particular it is possible to have self-similar cascades which do not show any fractal or multifractal behaviour when the box-counting analysis is performed. In the case of fractals the most relevant example is provided by processes having negative dimensions. With respect to multifractals, our results show that a strict interpretation of dissipation in a fully developed turbulent fluid as a result of a self-similar cascade is untenable.