Continuous eigenvalues of minimal subshifts via S-adic representations

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.

💡 Research Summary

This paper provides a comprehensive combinatorial description of continuous eigenvalues for minimal symbolic dynamical systems that admit an S‑adic representation satisfying natural mild conditions such as recognizability and primitivity. The authors work with a minimal subshift X equipped with a directive sequence τ = (τₙ)ₙ of morphisms (substitutions) between possibly varying alphabets. Under the basic hypotheses of primitivity and recognizability, Theorem 4.1 characterizes a real number α as a continuous eigenvalue precisely when there exists a sequence of real numbers ρ = (ρₙ)ₙ such that for every finite word u₀…u_k the quantities

 ‖ρₙ(u₀) − ρₙ(u_k) − α·(hₙ(u₀)+…+hₙ(u_{k−1}))‖₁

tend to zero as n→∞. Here hₙ(a) denotes the length of τ₀∘…∘τₙ₋₁(a), i.e., the height of the n‑th Kakutani‑Rohlin tower. The condition can be interpreted as a modular‑1 convergence of the “height‑weighted” sequence α·hₙ(a) after a suitable compensation by ρₙ. When the directive sequence is strongly primitive, Theorem 4.2 shows that the compensation ρₙ can be omitted: α·hₙ(a) itself converges modulo 1 to zero for each letter a.

The paper then refines these results under two additional structural assumptions: (i) the directive sequence has finite alphabet rank, or (ii) it is decisive (Definition 3.18). Decisiveness guarantees that the sequence of Kakutani‑Rohlin partitions generated by τ actually generates the topology of X, even when infinitely many alphabets are involved. Under either assumption, the compensating sequence ρₙ acquires a morphic structure and can be identified with a local letter‑coboundary (Definition 3.1). Theorem 5.2 and Theorem 5.3 give the precise correspondence: a continuous eigenvalue α exists if and only if there is a letter‑coboundary β such that α·hₙ(a) − β(a) converges to an integer for every a. This recovers and extends Host’s coboundary theory from the substitutive case to the full S‑adic setting, including non‑finitary sequences.

A central combinatorial tool introduced is the extension graph of the empty word, which records admissible left and right extensions of letters. The authors prove (Theorem 3.14) that the dimension of the real vector space of letter‑coboundaries equals r − 1, where r is the number of connected components of this graph. Consequently, the structure of the extension graph directly controls the possible continuous eigenvalues.

Section 6 explores the interplay between coboundaries, extension graphs, factor complexity p_X(n), and the rational dimension of the eigenvalue group. Lemma 6.5 gives an upper bound on the rational dimension of the eigenvalue group in terms of the growth of extension graphs and p_X(n). Lemma 6.7 provides an alternative proof of Tijdeman’s theorem on the minimal factor complexity of transitive subshifts with rationally independent letter frequencies, and recovers a result of Andrieu–Cassaigne concerning dendric subshifts. Lemma 6.2, together with Lemma 3.14 and Lemma 5.7, yields a new characterization of primitive substitutions that generate letter‑balanced subshifts, offering an alternative to Adamczewski’s criterion. Finally, Corollary 6.18 collects several sufficient conditions ensuring that all coboundaries are trivial, which in turn simplifies the eigenvalue description.

Section 7 presents a suite of examples illustrating the sharpness of the hypotheses. Example 7.1 shows that when the directive sequence is neither of finite alphabet rank nor decisive, a continuous eigenvalue may exist without any associated letter‑coboundary, demonstrating the necessity of the extra assumptions. Lemmas 7.4–7.6 contrast situations where coboundaries are non‑trivial yet do not correspond to eigenvalues, and vice versa. The final subsection treats constant‑length, finite‑rank S‑adic systems, proving via Lemma 4.1 that only rational additive eigenvalues can occur, a result previously obtained by different methods.

Overall, the paper establishes a robust combinatorial framework for continuous eigenvalues of minimal S‑adic subshifts. By linking modular convergence of tower heights, local coboundaries, and the topology of extension graphs, it unifies and extends earlier spectral results for substitutive systems, provides new bounds on eigenvalue groups, and supplies practical criteria for detecting or ruling out non‑trivial eigenvalues in a broad class of symbolic dynamical systems.