Before revision:  The core of this extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent  -- the fundamental exponent of the study -- and hence a multi-branched spectrum of dimensions , which for our case is additionally of the left-sided one. We examine the main thermodynamic consequences of the existence of this type of multifractality .
After revision:  The core of this extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent -- the fundamental exponent of the study -- and hence a multi-branched spectrum of dimensions . The Legendre-Fenchel transform is used instead of the routinely used canonical Legendre (single-branched) contact transform. Thermodynamic consequences of the existence of this type of multifractality .