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Let $f_k$, $k\\in\\{1,\\cdots,d\\}$, be $C^{1+\\omega_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $\\omega_1(t)\\cdots\\omega_d(t)=t\\omega(t)$ with $\\lim_{t\\rightarrow 0^+}\\omega(t)=0$, then $f_1,\\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1904.03984","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-04-08T12:09:13Z","cross_cats_sorted":[],"title_canon_sha256":"946e4249c336fc5dd0f84b1a0e95099da27f9ebec0d3ef3323dec6f9d182c289","abstract_canon_sha256":"5aad7a0184b8fd90f463249fa5d255d9e5866e76df2f42f7ecf352cd22f422e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:07.502056Z","signature_b64":"VEM5SnAtdbEZQxl65ow/sAjyjj6vbFnUw6sV5TyHMNvZUX3z+Jt+jRHtjP0/DA1uWmJpyfMarjICiBrbiqo/Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a8ddd069e951feaf42f826be1e1c30ec0ffbe53481322eaba4d9192c6f6e48b","last_reissued_at":"2026-05-17T23:49:07.501372Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:07.501372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commuting circle diffeomorphisms with their derivatives having mixed moduli of continuity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Enhui Shi, Hui Xu","submitted_at":"2019-04-08T12:09:13Z","abstract_excerpt":"Let $d\\geq 2$ be an integer and let $\\omega_1,\\cdots ,\\omega_d$ be moduli of continuity in a specified class which contains the moduli of H\\\"{o}lder continuity. Let $f_k$, $k\\in\\{1,\\cdots,d\\}$, be $C^{1+\\omega_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $\\omega_1(t)\\cdots\\omega_d(t)=t\\omega(t)$ with $\\lim_{t\\rightarrow 0^+}\\omega(t)=0$, then $f_1,\\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03984","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1904.03984","created_at":"2026-05-17T23:49:07.501474+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.03984v1","created_at":"2026-05-17T23:49:07.501474+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.03984","created_at":"2026-05-17T23:49:07.501474+00:00"},{"alias_kind":"pith_short_12","alias_value":"DKG52BU6SUP6","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"DKG52BU6SUP6V5BP","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"DKG52BU6","created_at":"2026-05-18T12:33:15.570797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3","json":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3.json","graph_json":"https://pith.science/api/pith-number/DKG52BU6SUP6V5BPQJV6DYODB3/graph.json","events_json":"https://pith.science/api/pith-number/DKG52BU6SUP6V5BPQJV6DYODB3/events.json","paper":"https://pith.science/paper/DKG52BU6"},"agent_actions":{"view_html":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3","download_json":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3.json","view_paper":"https://pith.science/paper/DKG52BU6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.03984&json=true","fetch_graph":"https://pith.science/api/pith-number/DKG52BU6SUP6V5BPQJV6DYODB3/graph.json","fetch_events":"https://pith.science/api/pith-number/DKG52BU6SUP6V5BPQJV6DYODB3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3/action/storage_attestation","attest_author":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3/action/author_attestation","sign_citation":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3/action/citation_signature","submit_replication":"https://pith.science/pith/DKG52BU6SUP6V5BPQJV6DYODB3/action/replication_record"}},"created_at":"2026-05-17T23:49:07.501474+00:00","updated_at":"2026-05-17T23:49:07.501474+00:00"}