Fractal Atomicity, a Fundamental Concept in the Dynamics of Complex Systems

Abstract

Applying a fractal method of analyzing the dynamics of the structural units of any complex system, a mathematical concept is built, namely that of fractal atomicity. The construction of such a concept involves defining dynamic variables in the form of fractal functions, defining scale resolutions, defining a principle of scale covariance as a fundamental principle of motion, equations of evolution, etc. Finally, some specific mathematical properties of the fractal atom are also established. IntroductionThe notion of non-atomicity for set functions plays a key role in Measure Theory and its applications and extensions. For classical measures taking values in finite dimensional Banach spaces, it guarantees the connectedness of range. Even just replacing σ-additivity with finite additivity for measures requires some stronger non-atomicity property for the same conclusion to hold.Because of their multiple applications in game theory or mathematical economics, the study concerning atoms and non-atomicity for additive, respectively, non-additive set functions has developed. Particularly, (non)atomic (purely) measures have been studied in different forms due to their special form and their special properties