Parametric homogeneity and non-classical self-similarity. I. Mathematical background

The concept of parametric homogeneity, which includes parametric homogeneous (PH) and parametric-quasi-homogeneous (PQH) functions, PH-sets and corresponding transformations, is a kind of non-classical self-similarity. Introduction and study of the concept is important for understanding scaling properties of natural phenomena. PH-functions and PQH-functions are natural generalisations of concepts of homogeneous and quasi-homogeneous functions when the discrete group of coordinate dilations is considered instead of the continuous group. Some properties of PH- and PQH-functions and some of their applications are studied. Several ways of constructing PH-function of arbitrary degree are described. It is shown that PH-functions often have fractal graphs and can be nowhere differentiable. Such common for fractal geometry objects as the von Koch curve, the Weierstrass-Mandelbrot and Takagi-Hopsen functions, the Canter staircase and set posses PH-properties. However, the fractal properties of an object and the property of parametric-homogeneity are independent of each other, and the considered scaling law (PH-law) is not a kind of fractal scaling. A nowhere differentiable self-similar function and some associated PH-functions are introduced as examples of non-fractal scaling.