Dice Question Streamline Icon: https://streamlinehq.com

Connected-component proliferation and unit fractal dimension in the cubic Fibonacci Hamiltonian

Ascertain whether the number of connected components of the spectrum of the cubic (three-dimensional) Fibonacci Hamiltonian Sp(H_λ^{(3)}) becomes infinite near the threshold λ ≈ 1 + √2, and determine whether there exists a range of λ for which Sp(H_λ^{(3)}) has infinitely many connected components while its fractal dimension equals 1.

Information Square Streamline Icon: https://streamlinehq.com

Background

Numerical experiments suggest a sharp structural change in the spectrum of H_λ{(3)} around λ ≈ 1 + √2, with a potential transition to infinitely many components even when the fractal dimension remains 1.

Confirming such a phenomenon would reveal a delicate interplay between topological complexity (gap proliferation) and fractal geometry (dimension) in higher-dimensional quasicrystal spectra.

References

Furthermore, we conjecture that the number of connected components of $Sp(H_{\lambda}{(3)})$ becomes infinite around $\lambda\approx1+\sqrt{2}$ and that there is a region where the number of connected components is infinite, but the fractal dimension is 1.

— Optimal Algorithms for Quantifying Spectral Size with Applications to Quasicrystals (2407.20353 - Colbrook et al., 29 Jul 2024) in Section 7: Final Remarks (Conjectures)