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Continuous eigenvalues of minimal subshifts via S-adic representations

Valérie Berthé, Paulina Cecchi-Bernales, Bastián Espinoza

TL;DR

This work develops a comprehensive framework to characterize continuous eigenvalues of minimal symbolic systems endowed with S-adic representations. By introducing local letter-coboundaries and extension graphs, the authors connect spectral data to detailed combinatorial and dynamical structures, deriving criteria that hold under recognizability and primitivity, and strengthening them with decisiveness or finite alphabet rank. They establish a duality between eigenvalues and measures through linear-algebraic constructions, obtain bounds on the rational dimension of eigenvalues, and apply these ideas to discrepancy, balance, and Tijdeman-type results. The results yield practical criteria for when eigenvalues arise and provide sharp examples illustrating the necessity of hypotheses and the limits of coboundary methods. Collectively, the paper advances the understanding of spectral properties in S-adic subshifts and broadens the toolkit for analyzing minimal symbolic systems via combinatorial and linear-algebraic methods.

Abstract

We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems described by $S$-adic structures satisfying natural mild conditions, such as recognizability and primitiveness. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations are expressed in terms of associated sequences of local coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between coboundaries and extension graphs, and we give several types of sufficient conditions for the nonexistence of trivial letter-coboundaries. As further results, we apply coboundaries in the context of bounded discrepancy, and in particular we obtain a simple characterization of letter-balance for primitive substitutive subshifts. Moreover, we recover a result of Tijdeman on the minimal factor complexity of transitive subshifts with rationally independent letter frequencies. Finally, we use linear-algebraic duality to refine known descriptions of the possible values of eigenvalues in terms of measures of bases.

Continuous eigenvalues of minimal subshifts via S-adic representations

TL;DR

This work develops a comprehensive framework to characterize continuous eigenvalues of minimal symbolic systems endowed with S-adic representations. By introducing local letter-coboundaries and extension graphs, the authors connect spectral data to detailed combinatorial and dynamical structures, deriving criteria that hold under recognizability and primitivity, and strengthening them with decisiveness or finite alphabet rank. They establish a duality between eigenvalues and measures through linear-algebraic constructions, obtain bounds on the rational dimension of eigenvalues, and apply these ideas to discrepancy, balance, and Tijdeman-type results. The results yield practical criteria for when eigenvalues arise and provide sharp examples illustrating the necessity of hypotheses and the limits of coboundary methods. Collectively, the paper advances the understanding of spectral properties in S-adic subshifts and broadens the toolkit for analyzing minimal symbolic systems via combinatorial and linear-algebraic methods.

Abstract

We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems described by -adic structures satisfying natural mild conditions, such as recognizability and primitiveness. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations are expressed in terms of associated sequences of local coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between coboundaries and extension graphs, and we give several types of sufficient conditions for the nonexistence of trivial letter-coboundaries. As further results, we apply coboundaries in the context of bounded discrepancy, and in particular we obtain a simple characterization of letter-balance for primitive substitutive subshifts. Moreover, we recover a result of Tijdeman on the minimal factor complexity of transitive subshifts with rationally independent letter frequencies. Finally, we use linear-algebraic duality to refine known descriptions of the possible values of eigenvalues in terms of measures of bases.
Paper Structure (33 sections, 51 theorems, 239 equations, 1 figure)

This paper contains 33 sections, 51 theorems, 239 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Topological dynamical systems
  5. Subshifts
  6. Morphisms and directive sequences
  7. Recognizability and Kakutani-Rohklin partitions
  8. Block presentations
  9. Coboundaries, extension graphs and decisiveness
  10. Letter-coboundaries and symbolic discrepancy
  11. Extension graphs
  12. Decisiveness
  13. General characterization of eigenvalues
  14. Statements
  15. General strategy
  16. ...and 18 more sections

Key Result

Lemma 2.2

Let $\boldsymbol\tau=(\tau_n:\mathcal{A}_{n+1}^*\to\mathcal{A}_n^*)_{n\ge0}$ be a primitive directive sequence. Then, there exists a sequence $(a_n \in \mathcal{A}_n)_{n\ge0}$ such that, for every $n\ge0$, the word $\tau_n(a_{n+1})$ starts with $a_n$.

Figures (1)

  • Figure 1: The extension graph $\Gamma_{X_{\sigma}}(\varepsilon)$ of $X_{\sigma}$ admits two connected components.

Theorems & Definitions (122)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 112 more