Complex multifractal measures and a generalized multifractal formalism

Strongly inhomogeneous multifractal measures involving extremely diverse local probability scales and exhibiting left-sided multifractal spectra are considered. The standard multifractal formalism is extended to such drastically inhomogeneous multifractal measures by introducing additional filtering variables and scaling exponents, conjugated to these variables. It is shown that the generalized multifractal formalism provides a more adequate and complete description of complex structures of strongly inhomogeneous multifractal measures, in comparison with conventional formalism.