Gap labels and asymptotic gap opening for full shifts

We discuss gap labeling 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 labeling theorem open simultaneously. Nevertheless, for a suitable choice of the single-site distribution, we show that for generic sampling functions, each 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.