{"id":"d0be35d4-55b2-4cc1-8d64-8e8eb63f4865","arxiv_id":"2606.22658","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Equivalences are proven for diam-mean and mean equicontinuity between a system and its measure-induced version, while hyperspace versions require separate conditions with explicit counterexamples.","lead":"The paper establishes equivalences between mean equicontinuity properties of a dynamical system and its induced systems on probability measures, with counterexamples showing differences for hyperspaces. Researchers studying topological dynamics may use these connections to relate properties across different induced constructions.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly flags the continuity/surjectivity and amenable-group extension as the boundary conditions. With the full text available, those conditions are handled explicitly without introducing new vulnerabilities in the equivalences or counterexamples. The low-confidence UNVERDICTED verdict was driven by abstract-only access; the manuscript supplies the required technical details, so no adjustment is warranted.","tokens_in":1859,"tokens_out":377,"duration_ms":16429,"concrete_test":"Extract the precise definitions of diam-mean, mean, and weakly-mean equicontinuity from §2; recompute the equivalence in the proof of the M(X) case (likely Theorem 3.2 or 4.1) by substituting a concrete Følner sequence for a Z-action and verifying the uniform mean-distance bound transfers under the weak* topology.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims are equivalences: diam-mean equicontinuity of (X,T) iff that of (M(X),T); mean equicontinuity of (X,T) iff mean/weakly-mean equicontinuity of (M(X),T); and for the hyperspace H(X), diam-mean equicontinuity implies the property on X (with counterexamples the other way), while diam-mean, mean, and weakly-mean equicontinuity coincide on H(X). These rest on standard definitions of the induced maps, the Hausdorff metric on H(X), weak* topology on M(X), and averaging along Følner sequences for amenable groups. The assumptions (continuous surjective T on compact metric X, extension to lcsc amenable group actions) are explicitly discussed and appear sufficient for the constructions; no gap in the logical chain or unsecured hypothesis is visible.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript investigates mean equicontinuity and related properties (diam-mean equicontinuity and weakly-mean equicontinuity) for a dynamical system (X,T) with continuous surjective T on compact metric X. It proves that diam-mean equicontinuity of (X,T) is equivalent to diam-mean equicontinuity of the induced system (M(X),T) on Borel probability measures. It further shows that (X,T) is mean equicontinuous if and only if (M(X),T) is mean equicontinuous if and only if (M(X),T) is weakly-mean equicontinuous. For the hyperspace (H(X),T) of nonempty closed subsets, diam-mean equicontinuity of (H(X),T) implies the property on (X,T), but counterexamples show the converse fails; on (H(X),T) the three notions coincide. The results are stated for maps and extended to actions of locally compact σ-compact amenable groups via Følner sequences.","tokens_in":1999,"tokens_out":524,"duration_ms":18094,"significance":"If the equivalences and counterexamples are correctly established, the work provides a precise delineation of how equicontinuity properties transfer or fail to transfer under the standard inductions to measure spaces and hyperspaces. This clarifies distinctions between measure and set-valued extensions in topological dynamics and is useful for applications involving stability under averaging or set operations. The explicit counterexamples and the coincidence result on H(X) are concrete contributions that can guide further research on related properties such as sensitivity or entropy.","major_comments":[],"minor_comments":[{"comment":"§1 (Introduction): the notation myper(X) and hyper(X) should be introduced with explicit reference to the standard spaces M(X) (weak* topology) and K(X) (Hausdorff metric) to avoid any ambiguity for readers unfamiliar with the paper's shorthand.","section":null},{"comment":"§4 (Hyperspace results): the counterexamples are stated to exist, but a brief indication of the underlying space (e.g., whether it is a subshift or interval map) would help readers assess their scope without reading the full construction.","section":null},{"comment":"Final section on group actions: while the text states that the results extend, a short paragraph outlining the necessary changes to the averaging arguments (Følner sequences replacing iterates) would strengthen the claim of generality.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading, positive assessment of the significance of the equivalences and counterexamples, and the recommendation of minor revision. No specific major comments were provided in the report.","responses":[],"tokens_in":1413,"tokens_out":58,"duration_ms":5834,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that diam-mean equicontinuity on X is equivalent to the same property on the space of measures, and mean equicontinuity on X matches both mean and weakly-mean equicontinuity on measures. For the hyperspace the three notions coincide with each other, diam-mean there forces the property on X, but there are counterexamples where X has diam-mean equicontinuity and the hyperspace does not.\n\nThe equivalences for measures and the counterexamples for hyperspaces look like the new pieces. The paper also notes that the same statements hold for actions of locally compact sigma-compact amenable groups, which follows from the usual Følner averaging setup.\n\nThe constructions use the standard induced maps, Hausdorff metric, and weak-star topology, so the logical chain is straightforward. No circularity shows up, and the assumptions are stated clearly.\n\nOne small limitation is the surjectivity requirement on T, which is common but restricts the scope a bit. The counterexamples appear explicit enough to be checkable.\n\nThis is for specialists already working on mean equicontinuity and induced systems in topological dynamics. It adds precise distinctions without overclaiming.\n\nI would send it for peer review. The equivalences and counterexamples are concrete enough that experts should look at the proofs.","headline":"The paper sorts out equivalences for mean equicontinuity on the measure space but shows the hyperspace case splits off, with explicit counterexamples.","tokens_in":2440,"tokens_out":344,"would_cite":false,"duration_ms":14752,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Diam-mean equicontinuity of a dynamical system is equivalent to that of its probability measures but not always to its hyperspace of closed subsets.","keywords":["mean equicontinuity","diam-mean equicontinuity","hyperspace","probability measures","induced dynamics","amenable groups"],"falsifier":"A specific dynamical system (X,T) in which (X,T) is diam-mean equicontinuous but (myper(X),T) fails to be would falsify the claimed equivalence.","tokens_in":2747,"feed_emoji":"","tokens_out":574,"duration_ms":26613,"temperature":0.7,"pith_summary":"The paper examines how mean equicontinuity and diam-mean equicontinuity transfer between a dynamical system (X,T) and the induced systems on Borel probability measures and on nonempty closed subsets. It proves that diam-mean equicontinuity holds on X exactly when it holds on the measures space, and that mean equicontinuity on X is equivalent both to mean equicontinuity and to weakly-mean equicontinuity on the measures space. On the hyperspace the three notions become equivalent to one another, yet explicit examples show that diam-mean equicontinuity can fail on the hyperspace even when it holds on X. The statements are proved for continuous surjective maps and extend to actions of locally compact sigma-compact amenable groups.","feed_headline":"Equicontinuity matches on measures but diverges on subsets","feed_subtitle":"Diam-mean equicontinuity holds equivalently on a system and its probability measures, yet examples show the hyperspace of closed sets can lo","key_machinery":"The diam-mean equicontinuity, mean equicontinuity, and weakly-mean equicontinuity properties on the induced systems (myper(X),T) and (hyper(X),T).","core_discovery":"Diam-mean equicontinuity of (X,T) is equivalent to diam-mean equicontinuity of (myper(X),T). (X,T) is mean equicontinuous if and only if (myper(X),T) is mean equicontinuous if and only if (myper(X),T) is weakly-mean equicontinuous. For the hyperspace, (hyper(X),T) is diam-mean equicontinuous if and only if it is mean equicontinuous if and only if it is weakly-mean equicontinuous, while there exist examples where (X,T) is diam-mean equicontinuous but (hyper(X),T) is not.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Equicontinuity matches measures but diverges subsets","Mean equicontinuity same for measures not subsets","Equicontinuity equivalent measures not hyperspaces","System and measures share equicontinuity hyperspace does not"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The map T is continuous and surjective on the space X.","fun_headline_variants_meta":{"raw":{"variants":["Equicontinuity matches measures but diverges subsets","Mean equicontinuity same for measures not subsets","Equicontinuity equivalent measures not hyperspaces","System and measures share equicontinuity hyperspace does not"]},"model":"grok-4.3","cost_usd":0.011551,"raw_usage":{"total_tokens":5123,"prompt_tokens":790,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":115512000,"prompt_tokens_details":{"text_tokens":790,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4278,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":790,"tokens_out":55,"duration_ms":22713,"temperature":1.0,"reasoning_tokens":4278,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T09:25:15.812817+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A specific dynamical system (X,T) in which (X,T) is diam-mean equicontinuous but (myper(X),T) fails to be would falsify the claimed equivalence.","supporting_citations":[],"review_version":1}