Zero Hausdorff dimension spectrum for the almost Mathieu operator

TL;DR

This paper proves that for a dense set of frequencies, the spectrum of the critical almost Mathieu operator has zero Hausdorff dimension, revealing intricate fractal properties at criticality.

Contribution

It establishes the existence of a dense G_delta set of frequencies with zero Hausdorff dimension spectrum for the critical almost Mathieu operator.

Findings

Spectrum has zero Hausdorff dimension for a dense G_delta set of frequencies.

Shows fractal nature of the spectrum at critical coupling.

Advances understanding of spectral properties in quasi-periodic operators.

Abstract

We study the almost Mathieu operator at critical coupling. We prove that there exists a dense $G_\delta$ set of frequencies for which the spectrum is of zero Hausdorff dimension.