Critical multifractality near monofractal states

It is shown, by comparison with laboratory and computer simulation data obtained on turbulent, percolation and growth processes, that global (i.e. for wide-range of the variable q) behavior of generalized dimension, Dq, of the random systems near monofractal (homogeneous) states is similar to a critical behavior of Dq in a vicinity of q = 0. It is also shown that adequate choice of the multifractal measure allows to detect the closeness of the random systems to their monofractal (homogeneous) states even for systems which seem to be far from monofractality (homogeneity). Application of this approach to multifractal analysis of some natural observations (earthquake sequences, luminosity-space galaxy distribution) is also briefly discussed