The Jacobs–Keane theorem from the $  \mathcal{S} $-adic viewpoint

Felipe Arbulú; Fabien Durand; Bastián Espinoza

doi:10.3934/dcds.2024052

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2024, Volume 44, Issue 10: 3077-3108. Doi: 10.3934/dcds.2024052

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The Jacobs–Keane theorem from the $ \mathcal{S} $-adic viewpoint

1.
U. de Picardie Jules Verne, LAMFA, CNRS-UMR 7352, F-80039 Amiens, France
2.
U. de Liège, Mathématiques discrètes, B-4000 Liège, Belgique

^*Corresponding author: Fabien Durand
^*Corresponding author: Fabien Durand

Received: July 2023

Revised: April 2024

Early access: April 28, 2024

Published online: April 28, 2024

The first author is supported by ANID (ex CONICYT) Doctorado Becas Chile 72210185 grant. The third author thanks Doctoral Fellowship CONICYT-PFCHA Doctorado Nacional 2020-21202229. All authors are supported by ANR-22-CE40-0011 project Inside Zero Entropy Systems.

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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.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 37B20, 37A05.

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