Spectra self-similarity for almost Mathieu operators

TL;DR

This paper investigates the self-similarity properties of the spectra of almost Mathieu operators, revealing their fractal structure and establishing mathematical connections with algebraic transformations and Morita equivalence.

Contribution

It numerically determines the self-similarity maps of the spectra and proves a continuity result, linking spectral self-similarity to algebraic and C*-algebraic structures.

Findings

Spectra exhibit fractal-like self-similarity maps.

Similarity maps involve algebraic and Mobius transformations.

Established a connection with Morita equivalence of rotation algebras.

Abstract

We determine numerically the self-similarity maps for spectra of the almost Mathieu operators, a two-dimensional fractal-like structure known as the Hofstadter butterfly. The similarity maps each have a horizontal component determined by certain algebraic maps, and vertical component determined by a Mobius transformation, indexed by a semigroup of the matrix group $GL_2(\Z)$. Based on the numerical evidence, we state and prove a continuity result for the similarity maps. We note a connection between the indexing of the similarity maps and Morita equivalence of rotation algebras $A_\theta$, a continuous field of C*-algebras.