Fractal Dimension of Fibonacci Word Attractors Tied to Subword Complexity — E8 Intelligence Research

FINDING: Subword complexity of Sturmian colorings on regular trees is minimal-unbounded, and generalized Fibonacci word fractals carry self-similar attractors whose Hausdorff dimension is governed by the word's recursive substitution — linking combinatorics on words to fractal geometry. | MATH: For Sturmian words, subword complexity \(p(n)=n+1\) (minimal unbounded). For Fibonacci word fractal, the Hausdorff dimension \(D\) satisfies \(D = \log(\varphi^2)/\log(\lambda)\) where \(\lambda\) is the contraction ratio of the iterated function system; for the standard Fibonacci word fractal, \(\lambda = 1/\varphi^2\) (since the substitution \(0\mapsto 01, 1\mapsto 0\) has dominant eigenvalue \(\varphi^2 = \varphi+1 \approx 2.618\)). The \(p\)-adic fractal string tube formula involves complex dimensions \(D + i n \frac{2\pi}{\log p}\) with oscillatory terms \(t^{D - i n 2\pi/\log p}\). | CONNECTION: The slope of the Sturmian word is \(1/\varphi^2 = (3-\sqrt{5})/2 \approx 0.382\) — the golden r

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com