Abstract: The heterogeneity h（ρt）and coarse-grained entropy S（ρtε）of the density distribution ρ in phase space and the derived computational expression of multiscale factor η* are used in the expansion of Weyl curvature into multiscaled curvature R*; and the multifractal in relation to the singular distribution of physical quantities on geometry is used in the preliminary establishment of the multifractal geometry of growing structures set in Riemann space; on this basis, the equation of non-conservative attractive field is set up by the introduction of the general dimension Du in relation to the fractal and informational and incident dimensions. The analysis here shows that the new result could render the suitable basis for the settlement of the questions such as the singularity of cosmology the unidentified matter in the cluster of galaxies, the doubt of information from black hole and the unification between gravitation theory and quantum physics, and so on.

Keywords: 
• gravitation /
• dissipative system /
• irreversibility /
• equation of attractive field /
• multifractal geometry