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If the Birkhoff normal form of the Hamiltonian at $\\cT_0$ satisfies a R\\\"ussmann transversality condition, the torus $\\cT_0$ is accumulated by KAM tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $\\o_0$.\n  For frequency vectors $\\o_0$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vec"},"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":"1311.7334","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-28T14:38:47Z","cross_cats_sorted":[],"title_canon_sha256":"13c3b4c6d28744542bfe4d8a28c9dcd3edb3faf896d8ca0e773c9aa5e862f219","abstract_canon_sha256":"a5b386ca0749bf46a76dd416e5f8ef982a877f9117a60a652fb592f4d8798c3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:15.070774Z","signature_b64":"/gM7iQe1893k0b7TLewXeegnjnURaLMQHR3JSyawqB7nh4ObCJsy4+hAT2cnzGaYZKAe+h+KDOEOkONnxiCUBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4b022c0d5eacca17110335a7580ed862ae5e96416844d9346f86216d5c88c1b","last_reissued_at":"2026-05-18T01:28:15.070087Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:15.070087Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Around the stability of KAM-tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Bassam Fayad, Hakan Eliasson, Rapha\\\"el Krikorian","submitted_at":"2013-11-28T14:38:47Z","abstract_excerpt":"We show that an analytic invariant torus $\\cT_0$ with Diophantine frequency $\\o_0$ is never isolated due to the following alternative. 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