A Fractal Eigenvector

TL;DR

This paper investigates the fractal structure of the eigenvector associated with the dominant eigenvalue of recursively constructed Mandelbrot matrices, linking eigenvalues to periodic orbits in the Mandelbrot set.

Contribution

It introduces a novel analysis of the eigenvector structure of Mandelbrot matrices and explores their fractal properties, connecting matrix eigenvalues to complex dynamics.

Findings

Eigenvector exhibits fractal-like structure

Dominant eigenvalue linked to periodic orbits

Analysis of singular vectors with less success

Abstract

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.