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REVIEW 3 major objections 6 minor 54 references

Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts

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Pith's one-line read Theorem 42: a shift map on a countable Markov shift extends continuously to its generalized Markov shift exactly when the candidate endomorphism $\alpha_0$ maps the commutative $C^*$-algebra $D_A$ into itself.

desk verdict Theorem 42 is a genuinely new characterization and the spectral-radius results are valuable, but Section 5 needs a real revision (full-measure justification and explicit unitality/compactness hypothesis) before I'd sign off. read the letter →

arxiv 2506.07487 v1 pith:BQ7DQYTG submitted 2025-06-09 math.DS math.FAmath.OA

Rodrigo Bissacot , Iván Diaz-Granados , Thiago Raszeja This is my paper
classification math.DSmath.FAmath.OA MSC 37B1046L0547A10
keywords generalizedMarkovshiftsExel-LacaalgebrascontinuousextensionoftheshiftmapweightedendomorphismsspectralradiusCuntz-Kriegerrenewalinvariantmeasures
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 asks when the shift map on a countable Markov shift $\Sigma_A$ can be extended continuously to the generalized Markov shift $X_A$, the compactification of $\Sigma_A$ that arises as the spectrum of the commutative $C^*$-algebra $D_A$ inside the Exel-Laca algebra. The central result, Theorem 42, is that such a continuous extension exists exactly when the candidate endomorphism $\alpha_0$ on the generators of $D_A$ maps $D_A$ into itself. The paper also isolates two structural classes of shifts with a continuous extension, the single empty-word class and the periodic renewal class, and gives explicit matrices where no extension exists. When the extension exists and the transition matrix is column-finite, the spectral radius of a weighted endomorphism $a\alpha$ is expressed as a variational maximum over invariant measures on $\Sigma_A\cup E_A$, extending the known formula for weighted endomorphisms from finite to countable alphabets.

What carries the argument

The central object is the generalized Markov shift $X_A$, the spectrum of the commutative $C^*$-subalgebra $D_A$ of the Exel-Laca algebra, whose points are configurations described by a stem (an infinite or finite admissible word) and, for finite stems, a root set of infinite emitters. The carrying object is the candidate endomorphism $\alpha_0(e_{\gamma,F})=e_{\omega_0\gamma,F}$ defined on generators, together with the Gelfand isomorphism $\pi:C_0(X_A)\to D_A$ that sends generalized cylinders to projections. The duality of Proposition 33, which turns an endomorphism of a commutative Banach algebra into a partial map on its spectrum, converts the algebraic extension problem into the topological problem of extending the shift, while the core-subalgebra extension results (Proposition 28 and Lemma 29) make the algebraic extension possible exactly when the image of $\alpha_0$ lies in $D_A$. For the spectral radius, the machinery is the variational formula of Theorem 65, applied to invariant measures on $\Sigma_A\cup E_A$.

What would settle it

One concrete check: construct a $\sigma$-invariant probability measure on $X_A$ assigning positive mass to a point of $Y_A\setminus E_A$. If such a measure exists, the maximum in (5.17) over $\Sigma_A\cup E_A$ misses its contribution and the formula is false. Another test case is the full-shift matrix of Example 54: sequences $\xi_n=n1^\infty$ and $\eta_n=n2^\infty$ both converge to the unique empty word, but their images under a continuous extension would have to converge to $1^\infty$ and $2^\infty$ respectively, so continuous extendability fails.

Watch

Extended reading notes

Core claim

The core claim is the equivalence in Theorem 42: for an irreducible transition matrix $A$ with unital Exel-Laca algebra, the following are equivalent: $\alpha_0(G_A)\subseteq D_A$; there is a $*$-endomorphism $\alpha$ of $D_A$ extending $\alpha_0$; and the shift map $\sigma$ extends continuously from $\Sigma_A\sqcup F_A$ to all of $X_A$. The proof uses the Gelfand transform to show that the candidate endomorphism becomes composition with $\sigma$ on the dense subset $\Sigma_A$, and uses core-subalgebra extension results to promote the algebraic map and to force, through duality, that the associated partial dynamics is defined everywhere. Proposition 68 then states that under column-finiteness and continuous extendability, $r(a\alpha)=\max_{\mu\in\mathrm{Inv}(\Sigma_A\cup E_A,\sigma)}\exp\int_{\Sigma_A\cup E_A}\ln|\hat a|\,d\mu$, and for weights supported on finitely many cylinders the maximum is attained on a finite-alphabet transitive subshift.

Load-bearing premise

The load-bearing premise is that every $\sigma$-invariant probability measure on $X_A$ gives full measure to $\Sigma_A\cup E_A$; this is asserted without proof in Proposition 68, and the spectral radius formula (5.17) depends on it directly.

Editorial extensions

If this is right

  • If Theorem 42 is correct, continuous extendability of the shift is equivalent to the algebraic condition $\alpha_0(G_A)\subseteq D_A$.
  • For column-finite matrices, Corollary 44 reduces the check to the empty-word generators $e_{e,F}$, so extendability can be decided from the limit behavior of the columns of $A$.
  • The single empty-word class and the periodic renewal class both admit continuous extensions; on the empty-word set $E_A$ the extension is a fixed point in the first class and a cycle in the second.
  • Proposition 68 and Theorem 71 give a variational formula for the spectral radius, and reduce the invariant-measure maximum to a finite-alphabet subshift when the weight is supported on finitely many cylinders.
  • For the renewal shift, Theorem 76 turns the spectral radius into an explicit product over cylinder partitions.

Reading between the lines

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

  • A direct consequence the authors do not spell out: for column-finite matrices, extendability becomes a finite inspection of column limits, so one can certify or refute continuous extension by checking the generators $e_{e,F}$ only.
  • The full-measure assertion behind Proposition 68 can be justified through Remark 14 by writing $Y_A=\bigcup_{n}\sigma^{-n}(E_A)$ and using invariance to force each preimage to have equal measure; adding this argument would close the only visible gap in the spectral radius formula.
  • The two extension classes suggest a testable pattern: when the column-limit points of an irreducible column-finite matrix form a finite periodic cycle, the shift should extend continuously with cyclic dynamics on the empty words, while infinitely many empty words (left open by the paper) may produce non-uniqueness or failure of extension.
  • If the variational formula extends beyond the column-finite case, it would connect the thermodynamic formalism on $X_A$ directly to the spectral theory of weighted endomorphisms; the paper only treats column-finite matrices.
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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

3 major / 6 minor

Summary. Summary: The paper studies generalized countable Markov shifts X_A associated to irreducible {0,1}-matrices A, viewed as the Gelfand spectrum of the commutative C*-subalgebra D_A of the Exel-Laca algebra. The central result (Theorem 42) equates (i) membership of α0(G_A) in D_A, (ii) existence of a *-endomorphism α on D_A extending the partially defined α0, and (iii) continuous extendability of the shift map σ from Σ_A ⊔ F_A to all of X_A. Section 5 applies the Kwaśniewski–Lebedev variational formula to obtain spectral radius formulas for weighted endomorphisms aα on column-finite shifts, including a renewal-shift specialization (Theorem 76).

Significance. Significance: If correct, Theorem 42 gives a complete operator-algebraic criterion for continuous extendability of the shift map, and the spectral radius formulas extend the work of Kwaśniewski and Lebedev to a class of non-locally-compact countable Markov shifts. The explicit Gelfand isomorphism of Theorem 30, the detailed structural results of Sections 2–4, and the large supply of examples (single empty-word and periodic renewal classes, counterexamples such as the full shift in Example 54 and Example 61) are valuable. The paper is clearly written and the main structural proofs are generally detailed. However, the spectral radius section contains several unproved or incorrectly stated steps, so the paper is not yet ready for publication in its current form.

major comments (3)
  1. [Section 5, Proposition 68(i), Eq. (5.17)] The formula (5.17) relies on the assertion that Σ_A ∪ E_A has full measure for every σ-invariant Borel probability measure on X_A. This assertion is stated without proof. The paper cites Remark 14 and ultimately [3, Proposition 54] for the decomposition Y_A = X_A \ Σ_A = ⋃_{n≥0} σ^{-n}(E_A), but it does not show that the transient set Y_A \ E_A is null for every invariant measure. A proof can be supplied by observing that points in Y_A \ E_A eventually enter E_A and never return to Y_A \ E_A, so Poincaré recurrence forces every invariant probability to give zero mass to Y_A \ E_A; this step must be included. Without it, the replacement of the integral over X_A by an integral over Σ_A ∪ E_A is not justified.
  2. [Section 5, application of Theorem 65] Theorem 65 is stated for a unital commutative C*-algebra and an endomorphism that preserves the unit. Proposition 68 assumes only that A is column-finite and that σ extends continuously to X_A, with no compactness or unitality hypothesis on X_A / D_A. If X_A is not compact, D_A is not unital and Theorem 65 does not apply directly. The authors should either add the assumption that OA is unital (equivalently X_A compact) throughout Section 5, or provide a unitization argument (for instance using ~D_A and the one-point compactification ~X_A) and verify that the spectral radius of aα is unchanged by this passage. As written, the proof of (5.17) and (5.18) rests on an inapplicable theorem for non-unital D_A.
  3. [Theorem 76, proof of Eq. (5.22)] The proof of Theorem 76 conflates standard cylinders [γ] ⊆ Σ_A with generalized cylinders C_γ ⊆ X_A. Equation (5.24) expresses the Gelfand transform of a in terms of the indicators 1_{C_{γ_k(n_k−1)}}, but the sets Z_x are defined using [γ_k(n_k−1)] and its complement. Standard cylinders contain only infinite sequences, whereas generalized cylinders also contain finite-word configurations. Consequently, the partition X_A = ⊔_x Z_x is false if [γ] denotes the standard cylinder: a finite-word configuration whose stem begins with γ_k(n_k−1) lies in C_{γ_k(n_k−1)} but not in [γ_k(n_k−1)], so it is assigned to the wrong cell Z_x. The subsequent claim that Z_x ⊆ [γ_k(n_k−1)] ⊆ Σ_A for x with x_k = 1 is therefore not valid in the intended interpretation. The proof must be rewritten using generalized cylinders throughout, and then it must explicitly handle the fact that cells Z_x with x ≠ 0 may intersect Y_A, using a measure argument analogous to the one missing from Proposition 68(i) to justify the integral decomposition.
minor comments (6)
  1. [Theorem 42, statement (3)] The phrase "σ : Σ_A ⊔ F_A → X_A is continuously extended in all X_A" should be rephrased as "σ has a continuous extension to all of X_A" for clarity.
  2. [Remark 45] The phrase "where the unity is preserved" should be "where the unit is preserved" or "where the endomorphism is unital."
  3. [Proposition 68(ii)] The decomposition a = Σ_{w∈W} λ_w e_w + Σ_{u∈U} λ_u e_u is not unique because the elements e_{γ,F} are linearly dependent (see Remark 74). The condition "λ_w = 0 for all w ∈ W" therefore depends on the chosen decomposition. The authors should clarify that the statement is independent of the decomposition, or fix a decomposition before stating the claim.
  4. [Figures 5, 8, 9] The captions contain the repeated phrase "where only the only" and should be corrected.
  5. [Section 4, Lemma 48] The notation in the proof is awkward: "for every p ∈ N, there exists N = N_p s.t. (ξ_{n_k})_i = 0 for every 1 ≤ i ≤ p" mixes the sequence index with the coordinate index. Please rewrite using clearer indexing (e.g., (ξ_n)_i for the i-th coordinate of ξ_n).
  6. [Theorem 76, notation] The use of [γ] for both standard and generalized cylinders is confusing. Since equation (5.24) uses C_γ, the sets Z_x should be defined consistently with that notation, or the text should explicitly state the convention.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; Theorem 42 is a genuine characterization. The spectral-radius section contains an unproved full-measure assertion, which is a proof gap rather than a circular reduction.

full rationale

The central result, Theorem 42, is not circular. The equivalence between (1) α0(GA) ⊆ DA and (3) continuous extendability of σ is proved directly: on the dense subset Σ_A, equation (3.12) identifies the Gelfand transform of α0(eγ,F) with 1_{Vγ,F}∘σ, so the inclusion α0(GA) ⊆ DA is equivalent to the continuous extendability of σ by explicit density and duality arguments. Neither condition is defined in terms of the other, and no fitted parameter or predicted quantity is involved. The proof of (2)⇒(3) uses the standard duality Proposition 33 of Kwaśniewski–Lebedev plus a clear contradiction argument, and (3)⇒(1) again uses density of Σ_A. The result therefore has independent mathematical content. The paper does rely heavily on the authors' earlier work [3] for the structure of the generalized Markov shift X_A, including the decomposition Y_A = ⋃_{n≥0} σ^{-n}(E_A) (Remark 14, citing [3, Proposition 54]), closure identities involving cylinders (Proposition 61 of [3]), and the stem/root description of configurations. This is substantial self-citation, but it is not circular: [3] is prior, independent work with external co-authors; its assumptions do not include Theorem 42 or the spectral-radius formula, and it provides structural inputs rather than the paper's conclusions. A genuine proof gap exists in Proposition 68(i): the proof says 'since the set Σ_A ∪ E_A has full measure for any invariant measure' without proving this. This assertion is load-bearing for equation (5.17), because it is exactly the step that restricts the maximum in Theorem 65 from invariant measures on X_A to invariant measures on Σ_A ∪ E_A. The assertion can likely be justified from Remark 14 and invariance/Poincaré recurrence, but as written it is omitted. This is a correctness risk, not a circularity: it does not reduce the formula to its own assumptions by definition, and it does not convert a fitted input into a prediction. Overall, the central operator-algebraic characterization is self-contained, and the cited prior work is independent support; the score of 2 reflects the heavy self-citation and the unproved full-measure assertion, not an actual circular derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's results rest on standard C*-algebra and dynamical system tools, plus the structural theory of generalized Markov shifts developed in the authors' companion work [3]. No new physical or mathematical entities are postulated; the single empty-word and periodic renewal classes are families of examples, not new objects. There are no fitted parameters.

assumptions (6)
  • domain assumption A is irreducible
    Standing assumption throughout; used to ensure topological transitivity and nonempty cylinders.
  • domain assumption The Exel-Laca algebra O_A is unital (in Theorem 42 and the spectral radius sections)
    Unitality is needed to apply the Kwasniewski-Lebedev theorem (Theorem 65) and to ensure D_A is unital. The non-unital case is only sketched in Remark 45.
  • domain assumption Structure of X_A, including Y_A = union over n of sigma^{-n}(E_A) and the description of E_A via limit points of columns
    Imported from the authors' earlier paper [3] (Propositions 46, 54, 61; Lemma 27). Used throughout, especially in the proof of Proposition 68.
  • standard math Kwasniewski-Lebedev theorem (Theorem 65) about spectral radius of weighted endomorphisms on unital commutative C*-algebras
    The central variational formula for the spectral radius is taken directly from [37].
  • standard math Core subalgebra extension result (Proposition 27 and Lemma 29)
    Used to extend the *-isomorphism between dense subalgebras to the C*-algebras in Theorem 30 and to extend alpha0 to alpha in Theorem 42.
  • domain assumption Column-finiteness of A in Sections 4 and 5
    Imposed in the single empty-word class, the periodic renewal class, and the spectral radius theorems. It ensures that certain sums in the definition of alpha0 are finite and that kappa(xi_n)_1 tends to infinity when kappa(xi_n)_0 does.

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Pith. "Pith review of Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts." pith.science (2026). https://pith.science/paper/BQ7DQYTG

@misc{pith2026250607487,
  author       = {Pith},
  title        = {Pith review of: Extendable Shift Maps and Weighted Endomorphisms on Generalized Countable Markov Shifts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQ7DQYTG}},
  note         = {Machine review of arXiv:2506.07487}
}
abstract

We obtain an operator algebraic characterization for when we can continuously extend the shift map from a standard countable Markov shift $\Sigma_A$ to its respective generalized countable Markov shift $X_A$ (a compactification of $\Sigma_A$). When the shift map is continuously extendable, we obtain explicit formulas for the spectral radius of weighted endomorphisms $a\alpha$, where $\alpha$ is dual to the shift map and conjugated to $\Theta(f)=f \circ \sigma$ on $C(X_A)$, extending a theorem of Kwa\'sniewski and Lebedev from finite to countable alphabets.

Figures

Figures reproduced from arXiv: 2506.07487 by the authors.

Figure 1
Figure 1. Forbidden and allowed fillings for configurations in Ωτ A via rule R3. (R4) given g ∈ FN and k ∈ N satisfying ξg = ξgk = 1, then ξgj−1 = 1 ⇐⇒ A(j, k) = 1, j ∈ N. Definition 13. We define the set FA of finite stem (finite word) configurations of XA that are not empty, and EA denotes the set of empty-stem (empty word) configurations. The shift map σ : ΣA ⊔ FA → XA is defined as (1.2) σ(ξ)g := ξκ(ξ) −1 0 g [PITH_FULL… view at source ↗
Figure 2
Figure 2. Fillings in Ωτ A via rule R4. Remark 14. In the definition above ΣA⊔FA is an open set of XA, and σ is a local homeomorphism. In particular, every element of YA is in the form σ −n (ξ 0 ), where ξ 0 ∈ EA and n ∈ N0 [3, Proposition 54]. In order to illustrate these configurations, we present some examples of XA for the renewal, pair renewal and prime renewal shifts. Later in this paper we will use them and other examp… view at source ↗
Figure 3
Figure 3. Empty words configurations of the pair renewal shift 2. Representations for DA. In this section, we present explicitly the Gelfand ∗-isomorphism between DA and C0(XA) for an arbitrary irreducible transition matrix A, by mapping the generators of each C ∗ -algebra. It is well-known that DA and C0(XA) are ∗-isomorphic, see for instance Theorem 8.4 of [16], and [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Description of the elements of Vβ,F . Given, for instance, β = β0β1β2 and F = {i, j}, i 6= j, the elements of Vβ,F are precisely the elements in XA whose the set of filled words include the picture above. that C(ΣA) for the finite alphabet case is a C∗ -algebra generat…
Figure 5
Figure 5. Figure 5: The continuously extended dynamics on XA for the renewal shift for the empty word element. The arrows point from an element of the space to its image under σ, where only the only. The elements presented above are written as an ordered pair stem-root. Now, we present an…
Figure 6
Figure 6. Figure 6: Typical matrix of a generalized Markov shift space that belongs to the periodic renewal class. (1) M has not zero columns; (2) |EA| = m ≤ ne < ∞; (3) A is column-finite. Proof. In fact, the compactness of XA implies that DA is unital, and so is OA. By Proposition 8.5 o…
Figure 7
Figure 7. Figure 7: Example of converging sequence of columns of a transition matrix A belonging to the periodic renewal class. (Q1 = 1). Item (iii) of Definition 56 is satisfied by construction, and there are only two infinite emitters, namely 1 and 2. Moreover, the matrix M from Definit…
Figure 8
Figure 8. Figure 8: The continuously extended dynamics on XA for the pair renewal shift for the empty word elements. The arrows point from an element of the space to its image under σ, where only the only. The elements presented above are written as an ordered pair stem-root. A generaliza…
Figure 9
Figure 9. Figure 9: The continuously extended dynamics on XA for the N-renewal shift for the empty word elements. The arrows point from an element of the space to its image under σ, where only the only. The elements presented above are written as an ordered pair stem-root. Theorem 65 ([37…

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