Before revision:  This kind of multifractality is a direct result of the nonmonotonic behavior of the generalized Hurst exponent and not because of its non-analytic behavior as it found before. We have examined the main thermodynamic consequences of the existence of this type of multifractality related to the thermal stable, metastable, and unstable phases within a hierarchy of fluctuations as well as to first and the second order phase transitions between them.
After revision:  This kind of multifractality is a direct result of the nonmonotonic behavior of the generalized Hurst exponent and not because of its non-analytic behavior as has been previously suggested. We examine the main thermodynamic consequences of the existence of this type of multifractality related to the thermal stable, metastable, and unstable phases within a hierarchy of fluctuations as well as to first and the second order phase transitions between them.