arXiv:2608.14175v1

Continuous pointwise ergodicity for semigroup actions on locally compact spaces

Raimundo Briceño, Godofredo Iommi

math.DSmath.FA37B0537A3047A3543A0746E1554H15

Paper metadata

Original submission: August 14, 2026
Claimed MSC: 37B05, 37A30, 47A35, 43A07, 46E15, 54H15
arXiv categories: math.DS, math.FA
Paper license: arXiv non-exclusive distribution license ↗
Comments: 40 pages

Abstract

We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without Følner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of Følner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.

AI-generated audit

Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The compact and locally compact characterizations of continuous pointwise ergodicity, the Følner-average equivalences for bicancellative left-amenable semigroups, and the quotient and proper-factor conclusions are correct under their stated existence and properness hypotheses.

Theorems 3.3 and 4.3Correct

Continuous pointwise ergodicity is equivalent to the mean-ergodic decompositions

Pages 15–16 and 20–21 · Theorems 3.3 and 4.3 · arXiv:2608.14175v1

In the compact case, continuity of the ergodic map produces a bounded projection $P$ by integration. Its range is exactly $\operatorname{Fix}(\kappa)$ and its kernel is the annihilator of all invariant measures, hence $\overline{\operatorname{Cob}(\kappa)}$ . Conversely, the mean-ergodic projection sends each Dirac mass to the unique invariant probability on its orbit closure. The one-point compactification correctly identifies escaping orbits with the fixed mass $\delta_\infty$ and converts weak* continuity plus vanishing at infinity into continuity of the compactified ergodic map, yielding the locally compact theorem.

Theorem 5.4Correct

Mean ergodicity, uniform Følner convergence, and weak* mean continuity are equivalent

Pages 24–26 · Theorem 5.4 · arXiv:2608.14175v1

Bicancellativity makes both left and right translates of a left Følner sequence left Følner. The proved Eberlein-type theorem identifies the closed coboundary space with functions whose averages converge uniformly to zero. This gives uniform convergence from the direct-sum decomposition and reconstructs the same projection from any convergent Følner scheme. Passing to adjoints gives the asserted weak* limits, while continuity on subprobabilities reconstructs a weak* continuous measure-valued map vanishing at infinity.

Theorem 6.5 and Proposition 6.7Correct

The compactified ergodic quotient and proper-factor results hold

Pages 29–31 · Theorem 6.5 and Proposition 6.7 · arXiv:2608.14175v1

The continuous compactified ergodic map has image exactly the compact set of ergodic invariant probabilities, so the induced quotient is homeomorphic to that set and the invariant-measure simplex is Bauer. Pullback identifies quotient functions with invariant functions, and annihilator duality gives the stated pushforward kernel. For a proper factor, compactification supplies a continuous equivariant surjection; lifting a measure and averaging a lift produces the unique invariant measure on the factor orbit closure. The quotient-map and compactification arguments then transfer continuity, vanishing at infinity, and uniformity.

02Proofs4 reported findingsCorrect

The proofs are correct and complete. The functional-analytic annihilator identities, measure-valued integration, compactification, Følner translation estimates, quotient duality, and factor-lifting arguments are used with the required hypotheses and with the correct weak, weak*, and norm topologies.

Proposition 2.1 and Theorem 2.3Correct and complete

The dual decompositions and measure-valued operator correspondence are valid

Pages 9–14 · Proposition 2.1 and Theorem 2.3 · arXiv:2608.14175v1

The pre-annihilator of dual coboundaries is the invariant-function space and the annihilator of primal coboundaries is the invariant-measure space. Closed-range properties of projections justify both direct-sum implications. For the map $\Psi$ , weak* continuity and vanishing at infinity give pointwise boundedness and hence a uniform total-variation bound; Gelfand integration then defines the adjoint operator. Conversely, a weak*-continuous adjoint evaluated on Dirac masses supplies both continuity and the vanishing condition.

Theorems 3.3 and 4.3Correct and complete

The projection and compactification proofs preserve all orbit and measure hypotheses

Pages 14–21 · Sections 3–4 · arXiv:2608.14175v1

Proposition 3.1 correctly proves that almost every point of an ergodic measure has orbit closure equal to its support, without assuming amenability. Krein–Milman then supplies the weak* span of ergodic measures used to identify the projection kernel. Properness is exactly what extends each action map to the one-point compactification, and a noncompact orbit closure has only $\delta_\infty$ as an invariant probability precisely when its original closure carries no invariant probability. These facts close every direction of the two characterization theorems.

Theorems 5.2 and 5.4Correct and complete

The Følner and mean-ergodic arguments are complete

Pages 21–26 · Section 5 · arXiv:2608.14175v1

Pointwise convergence to an invariant function first gives weak convergence by dominated convergence; Mazur's theorem and the left Følner estimates then upgrade it to norm convergence. Averages of elementary coboundaries converge uniformly to zero, and the converse follows because $f-A_Ff$ is a coboundary. Compact orbit closures provide tight empirical measures, whose subsequential limits are invariant. Alternating Følner sequences correctly show independence of the chosen sequence, and the adjoint calculations establish the dual limits.

Theorem 6.5 and Proposition 6.7Correct and complete

The quotient, disintegration, and factor proofs are complete

Pages 26–31 · Section 6 · arXiv:2608.14175v1

The compact quotient is Hausdorff because it is identified with the compact image of the continuous ergodic map. Invariant functions are constant on equal-measure fibers because the common invariant probability has nonempty support in both orbit closures. The pushforward kernel follows from the annihilator of the pullback range, and the bounded inverse on invariant measures is constructed from the mean-ergodic projection. In the factor proof, every probability on a compact factor orbit closure has a lift; a Følner cluster point of that lift is invariant and uniqueness upstairs forces the desired pushforward downstairs.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2608.14175v1
Authors listed: Raimundo Briceño, Godofredo Iommi
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v1
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Continuous pointwise ergodicity for semigroup actions on locally compact spaces

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Continuous pointwise ergodicity is equivalent to the mean-ergodic decompositions

Mean ergodicity, uniform Følner convergence, and weak* mean continuity are equivalent

The compactified ergodic quotient and proper-factor results hold

The dual decompositions and measure-valued operator correspondence are valid

The projection and compactification proofs preserve all orbit and measure hypotheses

The Følner and mean-ergodic arguments are complete

The quotient, disintegration, and factor proofs are complete

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match