Theoretical Bases of Identification of Solid Surface Fractality

The paper presents an analysis of the reliability of the method adopted in measuring techniques (particularly in AFM) for determining the fractal properties of rough surfaces. It is established that the parameter ? (0 < ? < 1) and, correspondingly, the Lip? class cannot be reliably defined using the finite part of the Fourier series and Lorentz theorem. Mandelbrot's definition and the Lorentz theorem are shown to be insufficient as a theoretical basis for defining fractality experimentally. As a result of the mixing of the two concepts of nonidentity forming the basis of the modern understanding of fractality, there appear inaccuracies. The first concept implies that the fractal function should be perceived as a "broken line", i.e., a nondifferentiable line. The second obligatory property of the fractal function is the geometrical self-similarity of the function at different scale levels. The examples cited in the paper indicate that these properties do not imply one another as they are different in their essence. A new method is proposed for identification of fractality in experimental research. It is based on checking the geometrical similarity of the additive components of the distribution function of asperity heights. This approach may serve as a basis for developing a new theory of the interactions of fractal rough surfaces.