Exactly self-similar left-sided multifractal measures

We introduce and investigate a family of exactly self-similar nonrandom fractal measures, each having stretched exponentially decreasing minimum probabilities. This implies that τ(q) is not defined for q<0 and that ${\mathit{q}}_{\mathrm{bottom}}$=0 is a critical value of q. Since the partition function does not scale for all values of q, these measures are not multifractals in the restricted sense due to Frisch and Parisi [in 2 Turbulence and Predictability of Geophysical Flows and Climate Dynamics, Proceedings of the Enrico Fermi International School of Physics, edited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, New York, 1985), p. 84] and to Halsey et al. [Phys. Rev. A 33, 1141 (1986)]. However, they are exactly self-similar, hence are multifractals in a much earlier and more general meaning of this notion [B. Mandelbrot, J. Fluid Mech. 62, 331 (1974)]. We show that in these measures the ‘‘free energy’’ τ(q) is singular at q=${\mathit{q}}_{\mathrm{bottom}}$, in the sense that τ(q)=-1+${\mathit{c}}_{\lambda }$${\mathit{q}}^{\lambda }$+${\mathit{c}}_1$q+${\mathit{c}}_2$${\mathit{q}}^2$+O(${\mathit{q}}^3$), where 0<λ is a ‘‘critical’’ exponent. For λ≤1, the transition in the f(α) is smooth (i.e., of infinite order), while for λ>1, the transition order is ≥2. We then use a new sampling method to study problems arising in the study of such transitions in case of undersampling.