The Jacobs--Keane theorem from the $\cS$-adic viewpoint

TL;DR

This paper explores the spectral properties of Toeplitz subshifts using the ${ m extbf{S}}$-adic framework, providing new characterizations of discrete spectrum and conditions for unique ergodicity.

Contribution

It introduces a novel ${ m extbf{S}}$-adic approach to analyze Toeplitz subshifts, extending classical results with new criteria based on coincidences density.

Findings

Characterizes spectral properties via coincidences density.

Provides necessary and sufficient conditions for unique ergodicity.

Revisits Jacobs--Keane theorem within the ${ m extbf{S}}$-adic framework.

Abstract

In the light of recent developments of the ${\mathcal S}$-adic study of subshifts, we revisit, within this framework, a well-known result on Toeplitz subshifts due to Jacobs--Keane giving a sufficient combinatorial condition to ensure discrete spectrum.We show that the notion of coincidences, originally introduced in the '$70$s for the study of the discrete spectrum of substitution subshifts, together with the ${\mathcal S}$-adic structure of the subshift allow to go deeper in the study of Toeplitz subshifts. We characterize spectral properties of the factor maps onto the maximal equicontinuous topological factors by means of coincidences density.We also provide an easy to check necessary and sufficient condition to ensure unique ergodicity for constant length ${\mathcal S}$-adic subshifts.