Coboundaries and eigenvalues of finitary $ S $-adic systems

Valérie Berthé; Paulina Cecchi Bernales; Reem Yassawi

doi:10.3934/jmd.2025004

Article Contents

2025, Volume 21: 271-325. Doi: 10.3934/jmd.2025004

This volume Previous Article Homologically non-trivial periodic orbits for generic Hamiltonian diffeomorphisms of surfaces Next Article Equidistribution of continued fraction convergents in

$ \mathrm{SL}(2,\mathbb{Z}_m) $

with an application to local discrepancy

Coboundaries and eigenvalues of finitary $ S $-adic systems

1.
Université de Paris, IRIF, CNRS, 8 Pl. Aurélie Nemours, F-75013 Paris, France
2.
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
3.
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom

Received: February 15, 2022

Revised: December 03, 2024

This work was supported by the Agence Nationale de la Recherche and the FWF Austrian Science Fund through the project "SymDynAr" (ANR-23-CE40-0024-01), and the EPSRC grants, numbers EP/V007459/2 and EP/S010335/1.
PCB: Supported by ANID/Fondecyt Postdoctorado 3210746, ANID/Fondecyt Iniciación 11240886, Vicerrectoría de Investigación y Desarrollo (VID)-Universidad de Chile/U-Inicia Project UI-003/23, ANID/STIC Amsud AMSUD230049 and by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of excellence from ANID-Chile.

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Abstract

An $ S $-adic system is a symbolic dynamical system generated by iterating an infinite sequence of substitutions or morphisms, called a directive sequence. A finitary $ S $-adic dynamical system is one where the directive sequence consists of morphisms selected from a finite set. We study eigenvalues and coboundaries for finitary recognizable $ S $-adic dynamical systems, i.e., those where points can be uniquely desubstituted using the given sequence of morphisms. To do this we identify the notions of straightness and essential words, and use them to define a coboundary, inspired by Host's formalism, which allows us to express necessary and sufficient conditions that a complex number must satisfy in order to be a continuous or measurable eigenvalue. We then apply our results to finitary directive sequences of substitutions of constant length, and show how to create constant-length $ S $-adic shifts with non-trivial coboundaries. We show that in this case all continuous eigenvalues are rational and we give a complete description of the rationals that can be an eigenvalue, indicating how this leads to a Cobham-style result for these systems.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 05A05.

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Figure 1. In this example, we construct $ \mathcal T_n(a) $ with $ \sigma_{n-1}(a) = cbd $, by concatenating portions of the towers of level $ n-1 $ for $ c $, $ b $ and then $ d $

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Figure 2. Part of the natural Bratteli diagram for a strongly straight directive sequence, where the letters $ a $ and $ b $ determine the same right-infinite limit word. Dashed edges correspond to minimal edges from $ a $ or $ b $ which lead directly to the limit word $ {\boldsymbol a} = {\boldsymbol b} $. The solid edges depict the limit word $ {\boldsymbol a} $

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