Novel Functional Representation of Fractal Functions

We propose a functional framework for the representation and construction of fractal 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. In contrast to classical approaches that define fractals through affine or nonlinear mappings acting on geometric sets, the present work formulates fractal generation explicitly in terms of functional operators. The proposed framework defines a class of recursively generated functions whose limiting behaviour exhibits self-similarity and fractal 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 fractal-like behavior 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 fractal structures, providing a foundation for further theoretical investigation and quantitative analysis.