Halsey’s Measure and Curdling Theorem

Abstract

For a multiplicative cascade (MC), we show that Halsey’s measure is non-degenerate when the partition function takes the value 1, it defines the function τ(q) and introduces a probability vector P(q) for MC. Using an Iterated Function System with Probability with its Hutchinson operator, we generate the MC, obtain the fractal decomposition of self-similar fractal F in subsets Jα, taking the statistical measure of these Jα with P(q) is showed that τ(q) and D(α) are a couple of Legendre transforms, and that for each value of q exist an statistical attractor Jα(q) called the curdling set, where is concentrated the statistical measure, this fact is denoted as the curdling theorem, using this theorem with the multifractal spectra is showed how is obtained the q value such Jα would be the curdling set of the MC.