Limit-Periodic Dirac Operators with Thin Spectra

TL;DR

This paper proves that limit-periodic Dirac operators typically have spectra of zero Lebesgue measure and zero Hausdorff dimension, using a novel commutation method that overcomes previous limitations in spectral analysis.

Contribution

It introduces a new, model-independent commutation argument to establish generic zero-measure spectra for limit-periodic Dirac and CMV operators.

Findings

Limit-periodic Dirac operators generically have zero Lebesgue measure spectra.

A dense set of these operators have spectra of zero Hausdorff dimension.

The new method applies to CMV matrices, showing zero-measure spectra in that setting.

Abstract

We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and that a dense set of them have spectra of zero Hausdorff dimension. The proof combines ideas of Avila from a Schr\"odinger setting with a new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, in Schr\"odinger-type settings, translation of the spectral measure corresponds to small $L^\infty$-perturbations of the operator data, but this is not true for Dirac or CMV operators. The new argument is much more model-independent. To demonstrate this, we also apply the argument to prove generic zero-measure spectrum for CMV matrices with limit-periodic Verblunsky coefficients.