We develop a generalized multifractal analysis based on the Legendre-Fenchel transform instead of the routinely used Legendre transform. We test this analysis for study time series consisting of interevent times. As a result, we have received the nonmonotonic behavior of the generalized Hurst exponent (which is a basic exponent herein) and hence a multi-branched left-sided spectrum of dimensions. 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.
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