Generative Mathematical Physics B: Continuization of Iterated Function Systems and Renormalization Group Flow: A Mathematical Isomorphism

Iterated function systems (IFS) and renormalization group (RG) flows, as twomathematical frameworks describing multiscale structures, have long been developed independently in fractal geometry and statistical physics, respectively. Thispaper establishes a rigorous mathematical isomorphism between them, proving thatunder specific conditions, the continuized parameter flow of an IFS is structurallyequivalent to the scale transformation of an RG flow. Through spectral analysis ofthe transfer operator and a fiber bundle construction on scale space, we demonstratethat the family of contraction maps of an IFS corresponds to the eigen-system of theJacobian matrix of an RG transformation, and the self-similarity equation of theIFS attractor maps to the scaling relations at an RG fixed point. This isomorphismtranslates geometric quantities such as fractal dimension and contraction ratio intophysical quantities like critical exponents and beta functions, providing a unifiedmathematical foundation for understanding the generative mechanisms of multiscale systems. Detailed theorem proofs and numerical verifications are presented,covering classical models such as the Cantor set and the Sierpiński gasket.