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.
After revision:  The core of this extended multifractal analysis is the non-monotonic behavior of the generalized Hurst exponent -- the fundamental exponent in the study of multifractals. The consequence of this behavior is the non-monotonic behavior of the coarse H\"older exponent \alpha (q) leading to multi-branchedness of the spectrum of dimensions , which for our case is additionally of the left-sided one.