Perron-Frobenius Theory: Spectral Backbone of Symbolic Dynamics and Self-Similar Tilings — E8 Intelligence Research

FINDING: Perron-Frobenius theory provides the spectral backbone for symbolic dynamics and substitution matrices, linking algebraic units to geometric scaling in self-similar tilings. | MATH: Perron-Frobenius theorem: for a nonnegative irreducible matrix \(A\), spectral radius \(ρ(A)\) is a simple eigenvalue with positive eigenvector; for primitive \(A\), \(limk→∞ A^k/ρ(A)^k = uv^T\) (rank-one projection). Substitution matrices \(M\) (e.g., Fibonacci substitution \(1↦ 12, 2↦ 1\) gives \(M={pmatrix}1&1\\1&0{pmatrix}\)) have PF eigenvalue \(ρ = {1+√5}{2} = φ ≈ 1.618\), with algebraic conjugate \(φ' = {1-√5}{2} ≈ -0.618\). Generalized eigenvalues of PF operators on symbolic spaces (arXiv:2106.12087) extend this to continuous spectra via analytic continuation of resolvents. | CONNECTION: The PF eigenvalue \(ρ\) for substitution matrices is precisely the inflation factor of the associ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com