Dimension-invariance, multifractality and complexity in some classical and quantum systems

Dimension-invariance approach, a dimension-stretching analogue of scale-stretching invariance, is developed for classical and quantum multifractal systems. It is shown, by comparison with laboratory and numerical experiments of different authors, that dimensions-invariance takes place in quantum pseudo-diffusion on lattice in systems with Hamiltonian having multifractal energy spectrum, in turbulence (near-critical generationog of the multiscaling states), for multifractal strange attractors, for mass multifractality in fragmentation, for the diffusion-limited corrosion, for energy dissipation in the fully developed turbulence, and for intermittent multiparticle production in hadron-nucleus interactions. Analogous results are also obtained for a large-scale structure of the universe (CfA catalog of Huchra's compilation of redshifts). Finally, an information complexity of a quantum system is briefly discussed and good correspondence with a recent numerical simulation is established.