Green-Function and Information-Geometric Correspondences Between Inverse Eigenvalue Loci of Generalized Lucas Sequences and the Mandelbrot Set

TL;DR

This paper explores numerical correspondences between inverse eigenvalue loci of generalized Lucas sequences and the Mandelbrot set, revealing geometric and potential-theoretic similarities across multiple scales.

Contribution

It introduces a multi-scale numerical framework for comparing algebraic spectral loci with nonlinear fractals, highlighting shared structural organization.

Findings

Spectral loci show low-distortion geometric correspondence with Mandelbrot boundary

Loci exhibit organization within the Mandelbrot Green function's external potential field

Multiple diagnostics confirm structural similarities beyond visual resemblance

Abstract

We investigate geometric, potential-theoretic, and information-theoretic correspondences between the inverse eigenvalue loci of companion matrices associated with generalized Lucas sequences and the boundary of the Mandelbrot set. Through systematic numerical experiments, we show that these algebraic spectral loci exhibit a striking low-distortion geometric correspondence with the Mandelbrot boundary at macroscopic scales, together with a coherent organization within its external potential field, characterized by concentration along narrow equipotential annuli of the Mandelbrot Green function. This correspondence is quantified using a suite of complementary diagnostics, including optimal transport matching, Procrustes alignment, local distortion measures, fractal and spectral statistics, Green-function-based potential comparisons, and convex simplex update analyses. Taken together, these results indicate that the observed similarity extends beyond visual resemblance, reflecting shared structural organization across geometric, harmonic, and statistical levels. While the present work is entirely numerical in nature, it establishes a robust multi-scale framework for comparing algebraic spectral constructions with nonlinear dynamical fractals, and it highlights several avenues for future analytical investigation.