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Homotopies and Maps between Eigenvalues of some Generalized Lucas Sequences and the Mandelbrot Set

Arturo Ortiz-Tapia

TL;DR

The paper investigates the relationship between the eigenvalues of companion matrices from generalized Lucas sequences and the Mandelbrot set \mathcal{M}. It develops statistical analyses of eigenvalue magnitudes, employs the Jungreis map for conformal uniformization, and constructs a family of homotopies (notably a sinusoidal one) that morph the eigenset \mathcal{L} toward the cardioid boundary of \mathcal{M}, with extensions via Douady’s tuning to baby Mandelbrot sets. A key contribution is a formal proof that the sinusoidal homotopy defines a homeomorphism (modulo countable exceptions) from \mathcal{L} to a dense subset of the cardioid boundary, and that composing with tuning extends this to stable components of \mathcal{M}. The work also classifies eigenvalues by dynamical type (Hyperbolic, Misurewicz, Parabolic, Siegel) to motivate dynamical-consistent refinements of homotopies, and proposes a broader program linking linear recurrences to nonlinear complex dynamics through topological and CW-complex frameworks. Together, these results establish a topological bridge between a countable spectral set and fractal regions of \mathcal{M}, suggesting principled ways to map linear-recursion spectra onto nonlinear parameter spaces with potential implications for spectral-dynamics correspondences.

Abstract

The eigenvalues of companion matrices associated with generalized Lucas sequences, denoted as L, exhibit a striking geometric resemblance to the Mandelbrot set M. This work investigates this connection by analyzing the statistical distribution of eigenvalues and constructing a variety of homotopies that map different regions of L to structurally corresponding subsets of M. In particular, we explore both global and piecewise homotopies, including a sinusoidal interpolation targeting the main cardioid and localized deformations aligned with the periodic bulbs. We also study a variation of the Jungreis map to better capture angular and radial structures. In addition to visual and geometric matching, we classify the eigenvalues according to their dynamical behavior, identifying subsets associated with hyperbolic, parabolic, Misurewicz, and Siegel disk points. Our findings suggest that meaningful correspondences between L and M must integrate both geometric deformation and dynamical classification. In light of these observations, we also suggest a conjectural homeomorphism between L and a dense subset of the Mandelbrot cardioid boundary, based on the behavior of the sinusoidal homotopy and the eigenvalue accumulation. Finally, we prove that the sinusoidal homotopy defines a homeomorphism (modulo countable exceptions) from L onto the boundary of the Mandelbrot cardioid, and, when composed with Douady's tuning map, extends to any baby cardioid or stable region of M, reinforcing the structural correspondence between these sets.

Homotopies and Maps between Eigenvalues of some Generalized Lucas Sequences and the Mandelbrot Set

TL;DR

The paper investigates the relationship between the eigenvalues of companion matrices from generalized Lucas sequences and the Mandelbrot set \mathcal{M}. It develops statistical analyses of eigenvalue magnitudes, employs the Jungreis map for conformal uniformization, and constructs a family of homotopies (notably a sinusoidal one) that morph the eigenset \mathcal{L} toward the cardioid boundary of \mathcal{M}, with extensions via Douady’s tuning to baby Mandelbrot sets. A key contribution is a formal proof that the sinusoidal homotopy defines a homeomorphism (modulo countable exceptions) from \mathcal{L} to a dense subset of the cardioid boundary, and that composing with tuning extends this to stable components of \mathcal{M}. The work also classifies eigenvalues by dynamical type (Hyperbolic, Misurewicz, Parabolic, Siegel) to motivate dynamical-consistent refinements of homotopies, and proposes a broader program linking linear recurrences to nonlinear complex dynamics through topological and CW-complex frameworks. Together, these results establish a topological bridge between a countable spectral set and fractal regions of \mathcal{M}, suggesting principled ways to map linear-recursion spectra onto nonlinear parameter spaces with potential implications for spectral-dynamics correspondences.

Abstract

The eigenvalues of companion matrices associated with generalized Lucas sequences, denoted as L, exhibit a striking geometric resemblance to the Mandelbrot set M. This work investigates this connection by analyzing the statistical distribution of eigenvalues and constructing a variety of homotopies that map different regions of L to structurally corresponding subsets of M. In particular, we explore both global and piecewise homotopies, including a sinusoidal interpolation targeting the main cardioid and localized deformations aligned with the periodic bulbs. We also study a variation of the Jungreis map to better capture angular and radial structures. In addition to visual and geometric matching, we classify the eigenvalues according to their dynamical behavior, identifying subsets associated with hyperbolic, parabolic, Misurewicz, and Siegel disk points. Our findings suggest that meaningful correspondences between L and M must integrate both geometric deformation and dynamical classification. In light of these observations, we also suggest a conjectural homeomorphism between L and a dense subset of the Mandelbrot cardioid boundary, based on the behavior of the sinusoidal homotopy and the eigenvalue accumulation. Finally, we prove that the sinusoidal homotopy defines a homeomorphism (modulo countable exceptions) from L onto the boundary of the Mandelbrot cardioid, and, when composed with Douady's tuning map, extends to any baby cardioid or stable region of M, reinforcing the structural correspondence between these sets.

Paper Structure

This paper contains 25 sections, 3 theorems, 45 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Statistics of |lambda|
  3. Matrix Construction
  4. The unit circle
  5. Uniformization and the Jungreis Map
  6. Numerical Implementation
  7. Behavior of Eigenvalues According to Their Modulus
  8. Homotopy from L to the cardioid of M
  9. A Homeomorphism between L and the Boundary of the Mandelbrot Cardioid
  10. Remark.
  11. Piecewise Homotopies
  12. Piecewise Scaling Transformation
  13. Homotopy Construction
  14. Snapshots of the Homotopy
  15. Observations
  16. ...and 10 more sections

Key Result

Lemma 1

The polynomials in Eq.Eq:CharacteristicPolynomials are irreducible in the rationals.

Figures (10)

  • Figure 1: A plot of $\mathcal{M}$ and the reciprocal of the eigenvalues coming from $A_{200}$.
  • Figure 2: Fit of the logistic equation to the data, with $\text{Model fit Parameters =}\{L\to94.4725,k\to0.0271419,x_0\to 111.11\}$.
  • Figure 3: Bin growth for interval (radius) 0.49--0.51.
  • Figure 4: Bin growth for interval (radius) 0.99--1.01.
  • Figure 5: Homotopy transformation for eigenvalues with $|\lambda| > 1$. The deformation is driven by the Jungreis coefficients.
  • ...and 5 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1: Homeomorphism between $\mathcal{L}$ and the Mandelbrot Cardioid Boundary
  • proof