REVIEW 2 major objections 3 minor 2 cited by
Extensibility and denseness of periodic semigroup actions
T0 review · 2 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash
Pith's one-line read Semigroup shift actions can inherit denseness of periodic measures from their free-group extensions.
desk verdict Solid semigroup extension of the periodic approximation property, but Theorem B as stated is false; the corrected closure version still supports the main applications. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the natural $G$-extension of an $S$-action: for an embeddable monoid $S$ and a receiving $S$-group $(G,\eta)$, this is the compact set of $G$-indexed tuples $(x_h)$ with $s\cdot x_h = x_{\eta(s)h}$, together with the projection $\pi$ onto $X$. For the shift action the natural extension is identified with $A^G$, and a measure is $G$-extensible when it is the pushforward of a $G$-invariant measure there. The bridge is Theorem B, which says that when the free $S$-group $\Gamma$ has the $(E)PA$ property, $\operatorname{Ext}_\Gamma(X,S)$ equals $P(X,S)$ (or $P_{\mathrm{erg}}$). In the free case the argument also uses Markov $\Sigma$-tree chains — measures defined by a positive root vector and stochastic matrices along edges of the Cayley tree — whose transition matrices are reflected across generators to build explicit $F_d$-extensions.
What would settle it
Take the free semigroup $F_2^+$ on two generators and the uniform Bernoulli measure on $\{0,1\}^{F_2^+}$; compute its natural pushforward to $\{0,1\}^{F_2}$ using the reflected Markov matrices from Proposition 4.10 and check $F_2$-invariance on the full two-sided shift. If any fully supported Markov measure on $A^{F_2^+}$ fails to extend to an $F_2$-invariant measure this way, the extensibility lemma behind Theorem D is wrong. Conversely, finding a left amenable residually finite semigroup whose free $S$-group is not amenable would break Corollary 4.4 and the EPA conclusion of Theorem C.
Extended reading notes
Core claim
The central discovery is that periodic measure denseness for a semigroup action can be reduced to the corresponding question for the free group on the semigroup, provided all invariant measures extend. The set of extensible measures $\operatorname{Ext}_\Gamma(X,S)$ — the image of the pushforward from $\Gamma$-invariant measures on the natural extension $X_\Gamma$ — is always weak-* closed, and when $\Gamma$ has the $(E)PA$ property it must coincide with the closure of the periodic measures $P(X,S)$. The paper also proves the two ingredients needed for the main applications: in the left amenable case the free $S$-group is the amenable group of right fractions, so a known specification-type theorem gives its EPA property; in the free case, every $S$-invariant measure on $A^S$ is extended to an $F_d$-invariant Markov tree chain on $A^{F_d}$. The same extensibility result characterizes when a receiving $S$-group is the free $S$-group, via Theorem E.
Load-bearing premise
The proof depends on previously established group-level results — a specification-type theorem for residually finite amenable groups and a theorem giving free groups the PA property — together with the companion paper's result that every surjective continuous S-action extends topologically to the free S-group; if any of these foundations gives way, the semigroup density theorems collapse.
Editorial extensions
If this is right
- Finitely supported invariant measures are weak-* dense in $M_S(A^S)$ for every left amenable semigroup $S$ that is residually a finite group, for every finite alphabet $A$.
- The same density holds for the free semigroup $F_d^+$ and for every subsemigroup $S = \langle\Sigma\rangle_+$ of $F_d$ with $\Sigma \supseteq \{a_1,\ldots,a_d\}$; in particular, periodic measures are dense on one-sided full shifts.
- In the left amenable case the density is ergodic: ergodic periodic measures, not just periodic ones, are dense.
- A semigroup with the PA property must be residually a finite group, so the paper's density results are optimal within the left reductive class.
- The free $S$-group is characterized measure-theoretically: it is the unique receiving $S$-group through which every fully supported Markov invariant measure extends.
Reading between the lines
- The same extensibility criterion could be used to hunt for further semigroup classes: any embeddable semigroup whose free $S$-group is residually finite and has the $(E)PA$ property, and whose shift measures are all extensible, will inherit the density property.
- The Markov tree reflection formula suggests a general construction: if a semigroup embeds in a group with a tree-like Cayley graph, one may be able to extend invariant measures by reflecting transition matrices along graph edges, yielding density results beyond free semigroups.
- Theorem E offers a practical algebraic test: a candidate receiving group for $S$ is the free $S$-group exactly when it extends every fully supported Markov measure on the full shift; a single non-extensible Markov measure disqualifies the candidate.
- Because finite orbits that are not completely $S$-invariant cannot support invariant measures, semigroup dynamics may need a distinct theory of 'eventual periodicity' to handle transient-but-finite behavior.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies periodic points and finitely supported invariant measures for continuous actions of countable semigroups. It introduces notions of pre-periodic and periodic points, proves a characterization of residual finiteness via denseness of periodic points in full shifts (Theorem A), and defines the (ergodic) periodic approximation property for semigroups. The central abstract result (Theorem B) relates the set of measures extensible to the free S-group with the finitely supported invariant measures. This is applied to show that left amenable semigroups that are residually finite groups have the EPA property (Theorem C) and that subsemigroups ⟨Σ⟩+ of free groups containing the positive generators have the PA property (Theorem D). A characterization of the free S-group in terms of measure extensibility is also given (Theorem E).
Significance. The paper provides a natural framework extending the group-level PA/EPA theory to semigroups, with a useful connection via natural extensions. Theorems C and D are genuinely new and non-trivial, and the proofs exhibit interesting techniques (Markov tree chains, Kolmogorov extension, free product constructions). However, Theorem B as stated is false; the '=' should be replaced by 'weak-* closure.' The applications remain valid under this correction, so the main positive results survive. The paper also relies on the companion preprint [6] for a key lemma, which should be made self-contained. With these revisions, the paper would be a solid contribution.
major comments (2)
- [§3.3, Theorem B] Theorem B as stated is false. The proof invokes Proposition 3.11(ii) to obtain Ext_Γ(X,S) ⊆ P(X,S), but that proposition has hypothesis P(X_G,G) = M_G(X_G), i.e., equality, not mere weak-* denseness. Since P(X,S) is not weak-* closed in general, the conclusion does not follow from denseness alone. For a concrete counterexample, take S = N, Γ = Z, X = A^N with |A| ≥ 2. Periodic measures are weak-* dense in M_Z(A^Z) for the full shift, and every N-invariant measure on A^N is Z-extensible via Kolmogorov extension, so Ext_Z(A^N,N) = M_N(A^N). Theorem B would then imply M_N(A^N) = P(A^N,N), contradicted by the non-finitely-supported Bernoulli(1/2) measure. The correct conclusion is Ext_Γ(X,S) = closure(P(X,S)) (and similarly with P_erg), obtained by combining Proposition 3.8, Proposition 3.11(i), continuity of π_*, and closedness of Ext. This correction still yields the 'In particular' clause, so the applications in Theorems C and D remain justified after adjustment. Please revise the statement and proof accordingly.
- [Appendix A, Lemma A.1] Theorem 4.5, which is essential for Theorem C, depends on Lemma A.1, stated there as '[6, Lemma 2.22]' without proof. Since this lemma is load-bearing and the companion preprint [6] is not yet published, please provide a self-contained proof or a detailed derivation, or at least state exactly which published source contains it. Similarly, Theorem 3.2 is quoted from [6]; while it is not used directly in the proofs of C and D, its role in the framework should be verifiable.
minor comments (3)
- [§3.3, Proof of Proposition 3.8] In the display '¯y = (yh)h∈G ∈ XΓΓΓ', the indexing set G should be Γ; the current notation is a typo.
- [§4.1, Remark 4.6] Remark 4.6 attributes a version of Theorem 4.5 to [15, Theorem 2.9], but reference [15] is 'Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem,' which appears unrelated; please verify the citation.
- [Abstract and Introduction] The phrase 'periodic semigroup actions' in the title and abstract is slightly misleading: the paper studies periodic points and measures for semigroup actions, not actions that are themselves periodic. Consider rephrasing for clarity.
Circularity Check
No significant circularity: semigroup density results are derived from external group-level theorems plus in-paper extension constructions; self-citations to the companion paper are parameter-free supporting lemmas, not assumed targets.
full rationale
We find no circular step in the claimed derivation. The target statements (Theorems C and D) are obtained by combining external, parameter-free inputs — Ren's theorem (Theorem 4.1), Bowen's theorem (Theorem 4.7), and Ore/Dubreil classical facts — with in-paper constructions: Proposition 3.8 embeds periodic measures into the natural extension, Proposition 3.11 pushes periodic measures down, Proposition 4.10/4.12 builds explicit F_d-extensions of Markov and then general invariant measures, and Appendix A constructs the extension measure for the reversible case. None of these steps assumes the conclusion that finitely supported invariant measures are weak-* dense in M_S(A^S); the density is obtained only after combining these inputs. The self-citations to [6] (Theorem 3.2 and Lemma A.1) are separate, parameter-free statements about topological extensibility and downward directedness of the preorder; they do not contain or presuppose the target density result, so under the stated rules they are independent supporting evidence and should not raise the circularity score. We do flag one non-circular correctness issue: in the proof of Theorem B (§3.3), Proposition 3.11(ii) is invoked with the hypothesis 'P(X_G,G) = M_G(X_G)', whereas Theorem B assumes only weak-* denseness. Denseness does not imply equality because P is not weak-* closed, so the step 'Ext_Γ(X,S) ⊆ P(X,S)' is not justified by the stated hypothesis, and the claimed equality Ext = P is at least not established by the given argument. This is an invalid inference in the derivation chain, not a self-referential reduction by construction.
Assumptions & free parameters
assumptions (8)
- standard math Every Borel probability measure on a compact metric space has a weak-* compact space M(X), and invariant measures form a closed convex set.
- standard math A countable group G is residually finite if and only if periodic points are dense in every full shift A^G (Proposition 1.6(iv), from [8]).
- domain assumption Every surjective continuous S-action is topologically G-extensible when G is the free S-group (Theorem 3.2, from [6]).
- domain assumption Residually finite amenable group actions with specification have dense periodic measures (Theorem 4.1, Ren [26]).
- domain assumption Free groups F_d have the PA property (Theorem 4.7, Bowen [5]).
- domain assumption A bicancellative left amenable semigroup is left reversible and its group of right fractions is amenable (Proposition 4.2 from [12] and [6]).
- domain assumption The preorder <=_S on the free S-group of a left reversible bicancellative semigroup is downward directed, equivalently gamma(S) is thick (Lemma A.1 from [6, Lemma 2.22]).
- standard math Markov tree chain measures exist and are unique, and S-invariance is characterized by the eigenvector and reversibility conditions (Proposition 4.9, from [17, p.240]).
Cite this review
Pith. "Pith review of Extensibility and denseness of periodic semigroup actions." pith.science (2026). https://pith.science/paper/EV65QSQL
@misc{pith2026250200312,
author = {Pith},
title = {Pith review of: Extensibility and denseness of periodic semigroup actions},
year = {2026},
howpublished = {\url{https://pith.science/paper/EV65QSQL}},
note = {Machine review of arXiv:2502.00312}
}
read the original abstract
We study periodic points and finitely supported invariant measures for continuous semigroup actions. Introducing suitable notions of periodicity in both topological and measure-theoretical contexts, we analyze the space of invariant Borel probability measures associated with these actions. For embeddable semigroups, we establish a direct relationship between the extensibility of invariant measures to the free group on the semigroup and the denseness of finitely supported invariant measures. Applying this framework to shift actions on the full shift, we prove that finitely supported invariant measures are dense for every left amenable semigroup that is residually a finite group and for every finite-rank free semigroup.
Forward citations
Cited by 2 Pith papers
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Natural extensions of embeddable semigroup actions
For an embeddable semigroup S, the free S-group is the unique group over which every surjective continuous S-action admits a natural extension, and left reversibility characterizes when all compact extensions factor t...
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Undecidability in the Ramsey theory of polynomial equations and Hilbert's tenth problem
Partition regularity of polynomial equations over Z is undecidable if Hilbert's tenth problem over Q is undecidable, and over function fields it is unconditionally Pi_2^0-complete.
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