Emergence of Quenched Phases and Second Order Transitions for Sums of Multifractal Measures

It is shown that superpositions of multifractal measures provide an ubiquitous mechanism for nonanalytic behavior of characteristic thermodynamic quantities. We find first and second order phase transitions. The latter frequently show up as experimentally observable stopping points in $f\left(\alpha \right)$ curves. Our results are derived analytically for sums of multiplicative and Markovian measures. The critical exponents of the continuous transition define a new universality class of systems, which include equivalent Ising models with long-ranged multispin interactions.