{"id":"07e4ca98-aa28-4a0a-9cda-addc59fa89da","arxiv_id":"2502.00312","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For embeddable semigroups, denseness of finitely supported invariant measures is governed by extensibility to the free group, and this yields the periodic approximation property for left amenable residually finite semigroups and finite-rank free subsemigroups.","lead":"This paper extends work on periodic points and periodic measures from group actions to semigroup actions, defining pre-periodic and periodic behavior for non-invertible dynamics. It proves that for left amenable semigroups that are residually finite groups, and for subsemigroups of free groups generated by subsets of the standard generators, finitely supported invariant measures are dense in the space of invariant measures on the full shift.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem B's proof replaces weak-* denseness by equality with the non-closed set P(X,S); as stated the theorem is false (e.g., S=N), though the corrected closure statement would support the applications.","rationale":"The paper's central framework is Theorem B, which is then used to derive the main semigroup PA/EPA results. The reader's verdict already identifies that Theorem B is false as stated, and the present stress test confirms this through a concrete instantiation. This is an internal logical error, not a disagreement with external consensus: the proof confuses weak-* denseness with equality for a non-closed set. The error is local and fixable; the corrected statement Ext_Γ(X,S) = closure(P(X,S)) is sufficient for the applications in Theorems C and D. The appendix's Kolmogorov-extension argument for left reversible semigroups is detailed and plausible, and the companion-preprint dependencies are a secondary risk. The reader's stated weakest_assumption field emphasizes external cited foundations, but the decisive load-bearing concern here is the internal Theorem B overclaim; hence partial agreement.","tokens_in":32550,"tokens_out":9969,"duration_ms":103097,"concrete_test":"Analytical check: instantiate Theorem B with S=N, free S-group Γ=Z, and X=A^N for a finite alphabet with |A| ≥ 2. Verify (1) P(A^Z,Z) is weak-* dense in M_Z(A^Z) (standard periodic-orbit density for the full shift); (2) every µ ∈ M_N(A^N) is Z-extensible via the two-sided Kolmogorov extension, so Ext_Z(A^N,N) = M_N(A^N); (3) the Bernoulli(1/2) product measure is N-invariant but not finitely supported. If Theorem B were true as stated, (1)–(2) would imply M_N(A^N) = P(A^N,N), contradicting (3). Replacing the conclusion by closure(P(X,S)) removes the contradiction and is exactly what the proof supports.","verdict_should_be":"UNCHANGED","load_bearing_attack":"At the final step of the proof of Theorem B (§3.3), the authors invoke Proposition 3.11(ii) to obtain Ext_Γ(X,S) ⊆ P(X,S). But Proposition 3.11(ii) is proved only under the hypothesis P(X_G,G) = M_G(X_G), whereas Theorem B assumes only weak-* denseness of P(X_G,G). The inference 'P is dense, Ext is closed, P ⊆ Ext, hence Ext = P' is invalid because P(X,S) is not weak-* closed: periodic measures can converge to non-periodic invariant measures. The argument actually yields Ext_Γ(X,S) = closure(P(X,S)), not equality with P(X,S). This is a genuine falsity, not a harmless strengthening. Take S = N, with free S-group Γ = Z, and X = A^N for a finite alphabet with |A| ≥ 2. Periodic measures are weak-* dense in M_Z(A^Z) for the full shift; every N-invariant measure on A^N extends to a Z-invariant measure on A^Z by Kolmogorov extension, so Ext_Z(A^N,N) = M_N(A^N). Theorem B would then force M_N(A^N) = P(A^N,N), contradicting the existence of the non-finitely-supported Bernoulli(1/2) measure.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":32740,"tokens_out":10450,"duration_ms":95840,"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":[{"comment":"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.","section":"§3.3, Theorem B"},{"comment":"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.","section":"Appendix A, Lemma A.1"}],"minor_comments":[{"comment":"In the display '¯y = (yh)h∈G ∈ XΓΓΓ', the indexing set G should be Γ; the current notation is a typo.","section":"§3.3, Proof of Proposition 3.8"},{"comment":"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.","section":"§4.1, Remark 4.6"},{"comment":"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.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real contribution to the semigroup side of periodic approximation, but Theorem B as printed is false. The error is localized and fixable, and the paper's main applications survive the correction.\n\nWhat is new: the semigroup notions of pre-periodic and periodic points (finite orbit plus complete invariance), the characterization in Theorem A linking residual finiteness of S to denseness of periodic points in full shifts, and the measure-theoretic framework connecting G-extensibility to periodic measures. Theorem C (left amenable + residually finite group => EPA) and Theorem D (subsemigroups of free groups generated by supersets of generators => PA) are the payoffs, and the proofs are mostly careful. The appendix's construction of the measure extension via Kolmogorov's theorem is solid, and the Markov tree chain argument in §4.2 is genuinely clever.\n\nThe soft spot: in §3.3, the proof of Theorem B invokes Proposition 3.11(ii) to get Ext_Γ(X,S) ⊆ P(X,S). But 3.11(ii) is only proved under the equality P(X_G,G)=M_G(X_G), not under weak-* denseness. Denseness plus closedness of Ext gives Ext = closure(P), not Ext = P, because P is not weak-* closed. The stress-test counterexample with S=N and the full N-shift over a two-symbol alphabet is correct: periodic measures are dense in M_Z, every N-invariant measure extends to a Z-invariant one, and the Bernoulli(1/2) measure is not finitely supported. So Theorem B as stated is false. That said, the corrected statement Ext = closure(P) is exactly what the applications in Theorems C and D need, since they conclude denseness. The framework stands, but the theorem needs rewording and the proof adjusted.\n\nOne more thing to watch: the paper leans on the authors' companion preprint [6] for Theorem 3.2 and Lemma A.1. Both are stated as separate results, and the appendix reproduces the relevant argument for 4.5, so this is not circular, but the new results inherit risk if [6] has issues.\n\nOverall: this is a paper worth engaging with. I would send it to a referee with instructions to verify the corrected Theorem B and check the inherited results from [6]. The team is doing serious work; the overclaim is a real flaw but not a fatal one.","headline":"Solid semigroup extension of the periodic approximation property, but Theorem B as stated is false; the corrected closure version still supports the main applications.","tokens_in":33334,"tokens_out":1635,"would_cite":true,"duration_ms":16323,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22D40","37A15","20M05","20M30","37B10","60J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Semigroup shift actions can inherit denseness of periodic measures from their free-group extensions.","keywords":["semigroup actions","periodic points","finitely supported invariant measures","natural extensions","free S-group","PA property","left amenable semigroups","Markov tree chains"],"falsifier":"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.","tokens_in":32295,"feed_emoji":"🔁","tokens_out":11821,"duration_ms":99800,"temperature":0.7,"pith_summary":"This paper extends the periodic approximation property (PA) and its ergodic version (EPA) from group actions to continuous semigroup actions. It introduces pre-periodic and periodic points for semigroups and proves that residual finiteness is equivalent to density of these points in full shifts. The central result is a bridge: for an embeddable semigroup $S$, if periodic measures are weak-* dense for the free $S$-group on the natural extension, then the $S$-periodic measures are exactly the $S$-invariant measures that extend to the free $S$-group. Using this, the paper proves that finitely supported invariant measures are weak-* dense for every left amenable semigroup that is residually a finite group, and for every subsemigroup of a free group generated by a set containing the generators.","feed_headline":"Periodic measures are dense for left-amenable and free semigroups","feed_subtitle":"Extensibility to the free group on the semigroup turns group density theorems into semigroup density theorems.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the topological natural extension theorem (every surjective continuous S-action extends to the free S-group) and the downward-directedness lemma used in Theorem 4.5","marker":"[6]"},{"why":"provides the theorem that residually finite amenable group actions with specification have dense ergodic periodic measures, giving the EPA property for the free S-group in the left amenable case","marker":"[26]"},{"why":"establishes that finitely generated free groups have the PA property, the group-level input for Theorem D","marker":"[5]"},{"why":"gives existence and universal property of the free S-group, the base object for all extensions","marker":"[10]"},{"why":"shows left amenable bicancellative semigroups are left reversible and have amenable group of right fractions, used in Corollary 4.4","marker":"[12]"},{"why":"the classical embedding theorem showing left reversible bicancellative semigroups are embeddable into a group of right fractions, identifying the free S-group in the amenable case","marker":"[25]"},{"why":"provides the Markov measure formalism on trees used to define Markov Sigma-tree chains and construct explicit F_d-extensions in the free case","marker":"[17]"}],"fun_headline_variants":["Periodic measures dense for left-amenable and free semigroups","Free group extension makes semigroup periodic measures dense","Dense periodic measures for semigroups via free group lift","Left-amenable and free semigroups: dense periodic invariant measures","Extensibility to free group yields dense periodic measures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Periodic measures dense for left-amenable and free semigroups","Free group extension makes semigroup periodic measures dense","Dense periodic measures for semigroups via free group lift","Left-amenable and free semigroups: dense periodic invariant measures","Extensibility to free group yields dense periodic measures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001283,"raw_usage":{"total_tokens":5199,"prompt_tokens":856,"completion_tokens":4343,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":4258}},"tokens_in":472,"tokens_out":4343,"duration_ms":31657,"temperature":1.0,"reasoning_tokens":4258,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T19:28:50.715324+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Natural extensions of embeddable semigroup actions","cited_arxiv_id":"2501.05536","evidence_quote":"supplies the topological natural extension theorem (every surjective continuous S-action extends to the free S-group) and the downward-directedness lemma used in Theorem 4.5"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the theorem that residually finite amenable group actions with specification have dense ergodic periodic measures, giving the EPA property for the free S-group in the left amenable case"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that finitely generated free groups have the PA property, the group-level input for Theorem D"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives existence and universal property of the free S-group, the base object for all extensions"},{"cited_title":"Donnelly","cited_arxiv_id":null,"evidence_quote":"shows left amenable bicancellative semigroups are left reversible and have amenable group of right fractions, used in Corollary 4.4"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"the classical embedding theorem showing left reversible bicancellative semigroups are embeddable into a group of right fractions, identifying the free S-group in the amenable case"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the Markov measure formalism on trees used to define Markov Sigma-tree chains and construct explicit F_d-extensions in the free case"}],"review_version":1}