Under the formalism of annealed averaging of the partition function, a type of random multifractal measures with the multipliers exponentially distributed is investigated in detail. Branching emerges in the curve of generalized dimensions, and negative values of generalized dimensions arise. Three equivalent methods of classification of the random multifractal measures are proposed, which is based on: (i) the properties of the curves of generalized dimensions, (ii) the solution properties of equation τ(q) = 0, and (iii) the relative position of the curve f(α) and the diagonal f(α) = α in the first quadrant. These classes of measures correspond to μ([0, 1]) = ∞, μ([0, 1]) = 1 and μ([0, 1]) = 0, respectively. Phase diagram is introduced to illustrate the diverse performance of the random measures that is multiplicatively generated. © 2001 Elsevier Science B.V.
Zhou, W. X., & Yu, Z. H. (2001). On the properties of random multiplicative measures with the multipliers exponentially distributed. Physica A: Statistical Mechanics and Its Applications, 294(3–4), 273–282. https://doi.org/10.1016/S0378-4371(01)00115-7