Novel Functional Representation of Fractal Functions

Abstract: We propose a functional framework for the representation and construction of self-similar structures based on recursive functional compositions, offering an alternative to purely geometric iterated function systems. We introduce a functional notion of self-similarity for real-valued functions, illustrated through log-periodic constructions, where complexity emerges under repeated scaling and composition. Unlike classical constructions, which characterise self-similar sets through affine or nonlinear transformations on geometric domains, the present work develops a formulation of self-similar generation in terms of functional operators acting on functions. The proposed framework defines a class of recursively generated functions whose limiting behaviour exhibits self-similar characteristics. Conditions under which these functions display irregular or jagged structures are discussed, including cases where smoothness is preserved while local complexity persists. Numerical experiments illustrate the emergence of self-similarlike behaviour under functional iteration, and a root-finding experiment highlights sensitivity to initial conditions consistent with chaotic dynamics. The results demonstrate that recursive functional constructions can serve as a viable analytical representation of self-similar structures, providing a foundation for further theoretical investigation and quantitative analysis.