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 (q) is not defined for q<0 and that qbottom=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 (q) is singular at q=qbottom, in the sense that (q)=-1+cq+c1q+c2q2+O(q3), where 0< is a critical exponent. For 1, the transition in the f() is smooth (i.e., of infinite order), while for >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.