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Continuous eigenvalues of minimal subshifts via S-adic representations and coboundaries

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Pith's one-line read Continuous eigenvalues of minimal S-adic subshifts are characterized by letter-coboundaries under decisiveness or bounded alphabet rank.

desk verdict Substantial S-adic eigenvalue paper with a genuine but repairable proof gap in Lemma 5.12; the central theorems are plausible and the new machinery is worth engaging. read the letter →

arxiv 2602.04833 v2 pith:QPNQRGAL submitted 2026-02-04 math.DS

Valérie Berthé , Paulina Cecchi-Bernales , Bastián Espinoza This is my paper · ORCID
classification math.DS MSC 37B1037A3068R15
keywords S-adicsubshiftscontinuouseigenvaluesletter-coboundariesextensiongraphsdecisivenesssymbolicdiscrepancybalancednessKakutani-Rohlintowers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper seeks to characterize exactly which real numbers are continuous eigenvalues of a minimal symbolic dynamical system when the system is presented as an S-adic subshift—an infinite composition of substitutions. The central answer is that, for a primitive recognizable S-adic structure that is either decisive or has bounded alphabets, a number α is an eigenvalue precisely when the tower heights of the Kakutani–Rohlin partitions approach α modulo 1 after being corrected by a letter-coboundary at every level. This matters because it turns an analytic spectral question into a finite combinatorial one: checking how height functions behave on the language of the subshift, or equivalently on return words. The same machinery yields eigenvalue groups expressed as integer combinations of tower-base measures, a criterion for letter-balanced substitutions, and rational-only eigenvalues for constant-length finite-rank directive sequences.

What carries the argument

The letter-coboundary is the central object: a morphism c:A*→R vanishing on every return word to a letter, equivalently satisfying c(a)=ρ(b)−ρ(a) whenever ab lies in the language. It converts the global eigenvalue equation into local linear conditions on adjacent letters. The extension graph Γ_X(ε), whose vertices are left and right copies of the alphabet and whose edges are length-2 words, encodes all letter-coboundaries: their real vector space has dimension r−1 for r connected components. Decisiveness is the third load-bearing tool: it says the first/last letter of τ_n(b) is constant on connected components of the level-(n+1) extension graph, which makes the Kakutani-Rohlin partitions gen

What would settle it

Take A={0,1} and alternate the substitutions τ0(0)=01, τ0(1)=00 and τ1(0)=10, τ1(1)=00 at every level; compute the first-letter maps f_{0,k}(a) = first letter of τ0∘τ1∘... applied to a. If these maps do not become eventually constant in k for both letters, Lemma 5.12's stabilization claim fails for a bounded-alphabet primitive recognizable sequence, and the proof of Theorems 5.2/5.7 must be repaired or replaced, even if the eigenvalue characterization itself survives.

Watch

Extended reading notes

Core claim

The paper's discovery is that the obstruction to a height sequence h_n converging to α modulo 1 is always a sequence of letter-coboundaries, provided the S-adic presentation is decisive or of finite alphabet rank. Letter-coboundaries are morphisms that vanish on return words, equivalently functions of the form c(a)=ρ(b)−ρ(a) on adjacent letters; they capture exactly the oscillatory corrections that cannot be removed by changing the approximating eigenfunction. Under decisiveness, Theorem 5.1 gives the equivalence sup_u ||c_n(u)−α h_n(u)||→0; under bounded alphabet rank, a single coboundary at one level suffices (Theorem 5.2), and under a strong local recurrence hypothesis a summable version

Load-bearing premise

The bounded-alphabet stabilization lemma—that after relabeling, the first-letter maps f_{ℓ,k}(a) eventually become constant in k for each level ℓ and letter a—is load-bearing; if it fails, the reduction of infinitely many level coboundaries to a single letter-coboundary in Theorems 5.2 and 5.7 collapses.

Editorial extensions

If this is right

  • For any minimal subshift with a primitive recognizable S-adic structure that is decisive or has bounded alphabet rank, eigenvalues are exactly integer linear combinations α=Σ w_a μ(B_n(a)) of tower-base measures for large n, refining the known inclusion E(X)⊆I(X).
  • Constant-length directive sequences of finite alphabet rank r have only rational eigenvalues, of the form p/(q|τ_{0,n}|) with 1≤q≤r.
  • A primitive aperiodic substitution generates a letter-balanced subshift exactly when its stable subspace plus its coboundary space has codimension 1, giving a new characterization of letter balance.
  • The extension-graph counting argument recovers the factor-complexity lower bound p_X(n) ≥ (n−1)(t−1)+|A| for transitive subshifts, with equality forcing all extension graphs to be trees.
  • Under any of the sufficient conditions (C1)-(C5), the eigenvalue criterion reduces to the classical tall-tower condition ||α h_n(u)||→0 and gives an eigenvalue representation as an integer vector on tower bases.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is an algorithmic eigenvalue sieve: for finite-rank S-adic systems, enumerate candidate values from integer combinations of tower-base measures and check the coboundary criterion on finitely many levels; the paper's examples suggest this is feasible even for non-proper substitutions.
  • The letter-coboundary dimension theorem suggests a conjugacy-independent spectral invariant: although the extension graph of the empty word is not conjugacy-invariant, the dimension of the coboundary space it controls may interact with stable subspaces in a way that gives balance and eigenvalue obstructions for morphic systems.
  • The constant-length result implies that any primitive recognizable constant-length S-adic subshift with an irrational continuous eigenvalue must have unbounded alphabet rank, a statement that can be checked against existing examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper develops S-adic characterizations of continuous additive eigenvalues for minimal subshifts. It introduces letter-coboundaries and a new 'decisiveness' condition, and proves in Section 4 general criteria for primitive recognizable directive sequences (Theorems 4.1--4.3). In Section 5, under decisiveness or bounded alphabets, these criteria are refined to local letter-coboundary conditions (Theorems 5.1--5.3), including a stabilized single-coboundary form for bounded alphabets. The paper also relates coboundaries to extension graphs (Theorem 3.14), derives a tower-measure duality for eigenvalues (Proposition 5.5), characterizes balancedness (Proposition 5.7), and applies the tools to balancedness, factor complexity, Tijdeman's theorem, and several worked examples, including a sharpness example for the necessity of the coboundary criteria.

Significance. If the results hold, this is a substantial extension of Host's coboundary approach to S-adic shifts, covering infinite alphabet rank under decisiveness and providing concrete, testable eigenvalue criteria. The paper is largely self-contained, with detailed proofs, a new structural notion (decisiveness), and useful connections between extension graphs and coboundaries. However, a load-bearing intermediate claim in the proof of Lemma 5.12 is currently false, and this blocks Theorems 5.2 and Proposition 5.7 as written. The likely fix is local, but it must be supplied before the central claims of Section 5 are established.

major comments (1)
  1. [Section 5.4, Lemma 5.12] The proof of Lemma 5.12 relies on the assertion that, for each fixed ell and a, the sequence (f_{ell,k}(a): k>=ell) of first-letter maps is eventually constant. This is false in general: the maps f_{ell,k}=f(tau_{ell,k}(.)) are products in the finite semigroup generated by a |-> f(tau_n(a)), and such products can be periodic. For example, take A={0,1}, tau_0(0)=10, tau_0(1)=01 (first-letter map is the swap), tau_1(0)=01, tau_1(1)=10 (first-letter map is the identity), and alternate. Then f_{0,k}(0) is 1 for odd k and 0 for even k, so it is not eventually constant. The subsequent construction of f_ell, f_infty, property (iii), and the invariance c = c o tau_{n_ell,n_k} used in (5.28) is therefore unsupported. This gap affects Theorem 5.2 and Proposition 5.7, both central claims. A diagonal/idempotent-selection argument over the finite semigroup may repair the lemma, but a complete proof m
minor comments (4)
  1. [Throughout] The internal numbering is inconsistent: statements labelled Theorem 5.1, Theorem 5.2, Theorem 5.3, Theorem 6.2 and Theorem 4.3 are subsequently referred to as 'Lemma 5.1', 'Lemma 5.2', 'Lemma 5.3', 'Lemma 6.2', and 'Lemma 4.3' in the proofs. This makes the paper difficult to read and should be harmonized.
  2. [Section 7.1, Eq. (7.4)] Equation (7.4), L_n(X_sigma x Y) = L_n(X_sigma) x L_{2^n}(Y), is dimensionally incorrect as written; a product subshift over A x A has length-n words with n letters in each coordinate. What is needed and what the construction actually gives is a relation involving L_{2^n}(X_sigma) x L_{2^n}(Y). This appears to be a typo rather than a structural gap, but it should be corrected.
  3. [Abstract] The abstract advertises an application to the Thue--Morse system in the rational base 3/2, but I could not locate this example in the body. Please either include the promised application or amend the abstract.
  4. [Section 2.3, Definition 2.1] The notation 'finite alphabet rank' is defined via a subsequence, while Theorem 5.2 and Proposition 5.7 assume bounded alphabets along all n. A sentence explaining that contraction allows passage between these assumptions would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the central characterizations are proved from standard external results and in-text lemmas, with only a non-circular proof gap flagged in Lemma 5.12.

full rationale

The derivation chain is self-contained. Theorem 4.1 is proved directly from the eigenfunction equation, uniform continuity, and Arzelà–Ascoli; Theorems 5.1 and 5.2 reduce that characterization to letter-coboundaries via Lemmas 5.8–5.10, whose proofs are included in the text. Lemma 3.14 is credited to [16] but is followed by a complete proof, so the self-citation is not load-bearing. External inputs (Mossé’s theorem, Gottschalk–Hedlund, Arzelà–Ascoli, Perron–Frobenius, Fitting lemma) are standard and independent. No fitted parameter is renamed as a prediction, no ansatz is imported via a self-citation, and decisiveness and letter-coboundaries are defined independently of the theorem conclusions. The reviewer’s concern about Lemma 5.12—specifically the assertion that “the sequence (f_{ℓ,k}(a):k≥ℓ) is eventually constant” for bounded alphabets—is a genuine correctness gap in the proof of Theorems 5.2/5.7, but it is not circularity: a broken or false intermediate step does not make the theorem’s statement an input to its own proof. The paper also frankly identifies recovered known results (e.g., Tijdeman’s theorem, [38, Theorem 2], [24]) rather than presenting them as new predictions, and these recoveries are not circular uses of the target results.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters; the characterizations are existence statements. The main new hypothesis, decisiveness, is a definition rather than an unproved entity. The broad applicability relies on an externally cited S-adic existence theorem, and the proofs use standard results from topological dynamics and linear algebra.

assumptions (4)
  • domain assumption Every infinite minimal subshift admits a primitive, recognizable, decisive S-adic expansion (Remark 3.20, citing [49] and [18]).
    This makes the decisive-sequence characterizations applicable to all infinite minimal subshifts, not just those already given an S-adic representation.
  • domain assumption Recognizability of primitive substitutions and S-adic sequences (Mossé's theorem [53] and its S-adic generalization [18]).
    Used throughout to form Kakutani–Rohlin partitions and to define unique τ-addresses.
  • standard math Gottschalk–Hedlund theorem (Theorem 3.7).
    Used to link bounded Birkhoff sums to real coboundaries, a key step in Sections 5 and 6.
  • standard math Arzelà–Ascoli theorem, Perron–Frobenius theorem, Fitting lemma, rank-nullity theorem.
    Used in the proofs of Lemma 4.1, Theorem 6.2, Lemma 6.4, and elsewhere; these are standard background results.

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Pith. "Pith review of Continuous eigenvalues of minimal subshifts via S-adic representations and coboundaries." pith.science (2026). https://pith.science/paper/QPNQRGAL

@misc{pith2026260204833,
  author       = {Pith},
  title        = {Pith review of: Continuous eigenvalues of minimal subshifts via S-adic representations and coboundaries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QPNQRGAL}},
  note         = {Machine review of arXiv:2602.04833}
}
abstract

We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems. These characterizations rely on a description of the system in terms of $S$-adic structures (i.e. infinite compositions of morphisms) satisfying natural mild conditions, such as recognizability and primitivity. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations involve sequences of letter coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between letter coboundaries and extension graphs, and we provide several sets of sufficient conditions ensuring the triviality of letter coboundaries. These results are applied, among other settings, to linear involutions and to the Thue--Morse system in the rational base $3/2$. We also illustrate the versatility of the notion of letter coboundaries in the context of bounded symbolic discrepancy. In particular, we recover a simple characterization of letter balance for primitive substitutive subshifts. Finally, we refine known descriptions of the possible continuous eigenvalues in terms of the measures of the bases of the towers provided by the $S$-adic representation.

Figures

Figures reproduced from arXiv: 2602.04833 by the authors.

Figure 1
Figure 1. The extension graph ΓXσ (ε) of Xσ admits two con￾nected components. any connected component K of ΓX(ε), that is, ρ is constant in the elements of the partition P R X. To state the result, we first introduce some notation. Let X ⊆ AZ be a minimal subshift. Denote by CX the set of letter-coboundaries c : A∗ → R in X. We let FX be the set of maps ρ: A → R that are constant in each C ∈ P R X (i.e., ρ(a) = ρ(b) for all a… view at source ↗

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.