Gap Labels and Asymptotic Gap Opening for Full Shifts

TL;DR

This paper investigates gap labelling for operators on full shifts, demonstrating conditions under which spectral gaps open or remain closed, especially in the large-coupling limit, and highlighting the influence of the single-site distribution.

Contribution

It provides new insights into gap opening phenomena for full shift operators, including generic conditions for spectral gaps to open in the large-coupling limit.

Findings

Spectral gaps open for generic sampling functions in the large-coupling limit.

Some gaps cannot open for purely diagonal operators with certain weights.

The gap labelling set is generated by weights of clopen subsets of the support.

Abstract

We discuss gap labelling for operators generated by the full shift over a compact subset of the real line. The set of Johnson--Schwartzman gap labels is the algebra generated by weights of clopen subsets of the support of the single-site distribution. Due to the presence of a dense set of periodic orbits, it is impossible to find a sampling function for which all gaps allowed by the gap labelling theorem open simultaneously. Nevertheless, for a suitable choice of the single-site distribution, we show that for generic sampling functions, every spectral gap opens in the large-coupling limit. Furthermore, we show that for other choices of weights there are gaps that cannot open for purely diagonal operators.