A Unified Framework for Fractal Geometry, Abstract Algebraic Structures, and Functional Analysis: Exploring Self-Similarity Through Measure Theory and Operator Algebras

This paper proposes a novel unified framework connecting fractal geometry with abstract algebraic structures and analytical methods. I develop a formalism that characterizes self-similar structures through the lens of group actions, measure theory, and operator algebras. My approach bridges previously disparate mathematical traditions, establishing formal connections between fractal dimension, algebraic invariants, and spectral properties of operators. I introduce several new theoretical constructs, including fractal homology groups, measure-preserving group actions on fractals, and spectral decomposition methods for self-similar operators. I demonstrate applications of this framework to physical systems with multiscale dynamics and biological pattern formation. The unification of algebraic and analytical perspectives offers new insights into the fundamental nature of selfsimilarity and creates opportunities for cross-disciplinary approaches to complex systems.