Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian

TL;DR

This paper investigates the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian, showing it equals the minimum of twice the dimension of the original spectrum and one for almost all couplings.

Contribution

It establishes a precise formula for the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian using dynamical systems and fractal geometry techniques.

Findings

Dimension equals min{2*HD_λ, 1} for almost all λ

Utilizes Fibonacci trace map dynamics and recent fractal sum dimension results

Provides a link between spectral theory and fractal geometry

Abstract

Denoting the Hausdorff dimension of the Fibonacci Hamiltonian with coupling $\lambda$ by $\mathrm{HD}_\lambda$, we prove that for all but countably many $\lambda$, the Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian with coupling $\lambda$ is $\min\{2\mathrm{HD}_\lambda, 1\}$. Our proof relies on the dynamics of the Fibonacci trace map in combination with the recent result of M. Hochman and P. Shmerkin on the Hausdorff dimension of sums of Cantor sets which are attractors of regular iterated function systems (Local entropy averages and projections of fractal measures, Ann. Math. 175 (2012), 1001--1059).