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We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ \\omega $. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary H\\\"{o}lder continuity with respect to parameter $"},"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":"2210.04383","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2022-10-10T01:05:35Z","cross_cats_sorted":[],"title_canon_sha256":"1bdb4244be71df923757a63db19eb8c977d8cb1e97f9ab535b75b2979647b9a4","abstract_canon_sha256":"74ccf1b30698d0925ad0ce08b5168c955c9336614c027ae21041ce602a7f1cee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:07:42.758239Z","signature_b64":"wJbQb4eeZjNdBevaT0NddfiT48Ck+duxGe5QafDQtz5yB5kssAmEpWgsGoQyyNwkxupvJyaeh2CIujDW3yHgCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2e3974624edf8612f7ec3d9ece51dfaa44e20ea561c5b9da6547fc4cffbef4c","last_reissued_at":"2026-07-05T09:07:42.757749Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:07:42.757749Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"KAM theorem on modulus of continuity about parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jiayin Du, Yong Li, Zhicheng Tong","submitted_at":"2022-10-10T01:05:35Z","abstract_excerpt":"In this paper, we study the Hamiltonian systems $ H\\left( {y,x,\\xi ,\\varepsilon } \\right) = \\left\\langle {\\omega \\left( \\xi \\right),y} \\right\\rangle + \\varepsilon P\\left( {y,x,\\xi ,\\varepsilon } \\right) $, where $ \\omega $ and $ P $ are continuous about $ \\xi $. 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