Green-function and information–geometric correspondences between inverse eigenvalue loci of generalized Lucas sequences and the Mandelbrot set
Year: 2026
Type: article
Abstract: The inverse eigenvalue loci of companion matrices associated with generalized Lucas sequences exhibit a striking large-scale geometric similarity to the Mandelbrot set [[EQUATION]]. Building on earlier homotopy constructions, this work develops a quantitative, multiscale comparison of their geometric, statistical, and information-theoretic structure. Using optimal-transport matching, rigid Procrustes alignment, and distortion and curvature diagnostics, it is shown that the inverse eigenvalue loci reproduce the macroscopic organization of [[EQUATION]] with uniformly small local distortion. These findings are consistent with a quasi-conformal correspondence at the level of large-scale envelopes, while allowing for elevated distortion near regions of high curvature or boundary transition.At a local analytic level, numerical conformal charts for the inverse eigenvalue loci are constructed via boundary-integral and finite-element solutions of Laplace problems, yielding stable Riemann-type mappings to the unit disk and the main cardioid. Local Cauchy–Riemann defects, Beltrami coefficients, and angle-distortion statistics indicate near-conformality away from the immediate boundary. Complementing these geometric results, a classical Kullback–Leibler contractive flow is applied to aligned histogram discretizations, producing a monotone decrease of divergence toward the Mandelbrot boundary distribution. Finally, a global potential-theoretic analysis based on the Mandelbrot Green function shows that multiple Lucas-type families concentrate on narrow equipotential annuli in the exterior of [[EQUATION]], defining stable spectral invariants. Together, these results establish a unified geometric–information correspondence between an algebraic spectral construction and a nonlinear dynamical fractal.
Source: Chaos Solitons & Fractals
Author: Arturo Ortiz–Tapia
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,970
OpenAlex ID: W7133928261