Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices

TL;DR

This paper extends gap labelling theory to ergodic Jacobi matrices, showing that the integrated density of states in spectral gaps belongs to a specific countable group related to the base dynamics.

Contribution

It establishes a Johnson-Schwartzman gap labelling theorem for ergodic Jacobi matrices, linking spectral gaps to the Schwartzman group of the underlying dynamics.

Findings

Proves the integrated density of states in spectral gaps belongs to the Schwartzman group.

Connects spectral properties of Jacobi matrices with topological invariants of the base dynamics.

Complements recent Johnson-type theorems for these matrix families.

Abstract

We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.