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 \(\rho(A)\) is a simple eigenvalue with positive eigenvector; for primitive \(A\), \(\lim_{k\to\infty} A^k/\rho(A)^k = \mathbf{u}\mathbf{v}^T\) (rank-one projection). Substitution matrices \(M\) (e.g., Fibonacci substitution \(1\mapsto 12, 2\mapsto 1\) gives \(M=\begin{pmatrix}1&1\\1&0\end{pmatrix}\)) have PF eigenvalue \(\rho = \frac{1+\sqrt{5}}{2} = \varphi \approx 1.618\), with algebraic conjugate \(\varphi' = \frac{1-\sqrt{5}}{2} \approx -0.618\). 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 \(\rho\) 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