Spectral Homogeneity of Discrete One-Dimensional Limit-Periodic Operators

TL;DR

This paper demonstrates that a dense set of limit-periodic operators in one dimension have spectra that are homogeneous Cantor sets, with some exhibiting purely absolutely continuous spectrum, using Floquet theory and dynamical formalism.

Contribution

It introduces a method to construct limit-periodic operators with homogeneous Cantor spectra and extends results to p-adic hulls using dynamical systems techniques.

Findings

Dense subset of limit periodic operators have homogeneous Cantor spectra.

Some operators exhibit purely absolutely continuous spectrum.

The construction applies to arbitrary p-adic hulls.

Abstract

We prove that a dense subset of limit periodic operators have spectra which are homogeneous Cantor sets in the sense of Carleson. Moreover, by using work of Egorova, our examples have purely absolutely continuous spectrum. The construction is robust enough to extend the results to arbitrary p-adic hulls by using the dynamical formalism proposed by Avila. The approach uses Floquet theory to break up the spectra of periodic approximants in a carefully controlled manner to produce Cantor spectrum and to establish the lower bounds needed to prove homogeneity.