A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices \(M_n\) for \(n=1\), \(2\), \(\ldots\) have nonnegative entries (indeed just \(0\) and \(1\), so each \(M_n\) can be called a binary matrix) and have eigenvalues whose negatives \(-\lambda = c\) give periodic orbits under the Mandelbrot iteration, namely \(z_k = z_{k-1}^2+c\) with \(z_0=0\), and are thus contained in the Mandelbrot set. By the Perron--Frobenius theorem, the matrices \(M_n\) have a dominant real positive eigenvalue, which we call \(\rho_n\). This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and similarly examines (with less success) the dominant singular vectors of \(M_n\) from the singular value decomposition.