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In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$ approximated by one with a positive Lyapunov exponent for a positive-measured subset of $M$. That is, the set \\[ \\left\\{ f\\in \\mathcal{S}ym^1_{\\omega}(M)\\,| \\begin{array}{ll} &\\mbox{The largest Lyapunov exponent }\\lambda_1(f,\\,x)>0\\\\ &\\mbox{ for a positive measure set } \\end{array} \\right\\} \\] is dense in $\\mathcal{S}ym^1_{\\omega}(M)$. \\end{abstract} \\end{center"},"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":"1506.05181","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","primary_cat":"math.DS","submitted_at":"2015-06-17T01:45:36Z","cross_cats_sorted":[],"title_canon_sha256":"66b11f6d672ee2d992ac7c808a35ac5da15e3ac0221a5b4ae7f9dc9f8cdfbf73","abstract_canon_sha256":"01c5765b041314fd80d95844f61fc3e476b9c5c75ea6c7834969f783aab33cc2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:46:12.852300Z","signature_b64":"rSV5xXPrnKcwfGdtru5Cq8mwaQCLO4UIvrB6d11jms4AtA1S1I056tCsL8s4plziFuY8KCNPB4gjbDgnUu7eAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7b3558a7d3849426a65b4cdc49fa8435bf9ebcb36f8511eaad2885776ceb219","last_reissued_at":"2026-05-18T01:46:12.851710Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:46:12.851710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On density of positive Lyapunov exponents for $C^1$ symplectic diffeomorphisms","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Chao Liang","submitted_at":"2015-06-17T01:45:36Z","abstract_excerpt":"Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $\\omega$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$ approximated by one with a positive Lyapunov exponent for a positive-measured subset of $M$. That is, the set \\[ \\left\\{ f\\in \\mathcal{S}ym^1_{\\omega}(M)\\,| \\begin{array}{ll} &\\mbox{The largest Lyapunov exponent }\\lambda_1(f,\\,x)>0\\\\ &\\mbox{ for a positive measure set } \\end{array} \\right\\} \\] is dense in $\\mathcal{S}ym^1_{\\omega}(M)$. \\end{abstract} \\end{center"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.05181","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":"1506.05181","created_at":"2026-05-18T01:46:12.851812+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.05181v1","created_at":"2026-05-18T01:46:12.851812+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.05181","created_at":"2026-05-18T01:46:12.851812+00:00"},{"alias_kind":"pith_short_12","alias_value":"46ZVLCT5HBEU","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"46ZVLCT5HBEUE2TF","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"46ZVLCT5","created_at":"2026-05-18T12:29:05.191682+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/46ZVLCT5HBEUE2TFWTG4JH5IIN","json":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN.json","graph_json":"https://pith.science/api/pith-number/46ZVLCT5HBEUE2TFWTG4JH5IIN/graph.json","events_json":"https://pith.science/api/pith-number/46ZVLCT5HBEUE2TFWTG4JH5IIN/events.json","paper":"https://pith.science/paper/46ZVLCT5"},"agent_actions":{"view_html":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN","download_json":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN.json","view_paper":"https://pith.science/paper/46ZVLCT5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.05181&json=true","fetch_graph":"https://pith.science/api/pith-number/46ZVLCT5HBEUE2TFWTG4JH5IIN/graph.json","fetch_events":"https://pith.science/api/pith-number/46ZVLCT5HBEUE2TFWTG4JH5IIN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN/action/storage_attestation","attest_author":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN/action/author_attestation","sign_citation":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN/action/citation_signature","submit_replication":"https://pith.science/pith/46ZVLCT5HBEUE2TFWTG4JH5IIN/action/replication_record"}},"created_at":"2026-05-18T01:46:12.851812+00:00","updated_at":"2026-05-18T01:46:12.851812+00:00"}