Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

TL;DR

This paper introduces a class of subshifts called leading sequences, proves uniformity of cocycles over them, and derives spectral properties of associated operators, with applications to Schreier graphs of groups acting on trees.

Contribution

It establishes the uniformity of locally constant cocycles over leading sequence subshifts and derives spectral results for related operators, covering simple Toeplitz and Sturmian subshifts.

Findings

Cocycle uniformity over leading sequence subshifts

Cantor spectrum of Lebesgue measure zero for associated Jacobi operators

Applicability to spectral theory of Schreier graphs for groups on rooted trees

Abstract

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.