Scaling of mathematical fractals and box-counting quasi-measure

The same term, ‘fractals’ incorporates two rather different meanings and it is convenient to split the term into physical or empirical fractals and mathematical ones. The former term is used when one considers real world or numerically simulated objects exhibiting a particular kind of scaling that is the so-called fractal behaviour, in a bounded range of scales between upper and lower cutoffs. The latter term means sets having non-integer fractal dimensions. Mathematical fractals are often used as models for physical fractal objects. Scaling of mathematical fractals is considered using the Barenblatt–Borodich approach that refers physical quantities to a unit of the fractal measure of the set. To give a rigorous treatment of the fractal measure notion and to develop the approach, the concepts of upper and lower box-counting quasi-measures are presented. Scaling properties of the quasi-measures are studied. As examples of possible applications of the approach, scaling properties of the problems of fractal cracking and adsorption of various substances to fractal rough surfaces are discussed.