{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:5IWWT5HMC343LEDYSROYDZKWFD","short_pith_number":"pith:5IWWT5HM","schema_version":"1.0","canonical_sha256":"ea2d69f4ec16f9b59078945d81e55628f453ce524b771d11082c9c2f85a6a5d8","source":{"kind":"arxiv","id":"1108.3747","version":1},"attestation_state":"computed","paper":{"title":"H\\\"older continuity of Lyapunov exponent for quasi-periodic Jacobi operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Kai Tao","submitted_at":"2011-08-18T13:06:16Z","abstract_excerpt":"We consider the quasi-periodic Jacobi operator $H_{x,\\omega}$ in $l^2(\\mathbb{Z})$ $(H_{x,\\omega}\\phi)(n) = -b(x+(n+1)\\omega)\\phi(n+1) - b(x+n\\omega)\\phi(n-1) + a(x+n\\omega)\\phi(n) = E\\phi(n),\\ n\\in\\mathbb{Z},$ where $a(x),\\ b(x)$ are analytic function on $\\mathbb{T}$, $b$ is not identically zero, and $\\omega$ obeys some strong Diophantine condition.\n  We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $\\rho_0=\\rho_0(a,b,\\omega,E_0)$, $\\beta=\\beta(a,b,\\omega)$ such that $|L(E)-L(E')|<|E-"},"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":"1108.3747","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2011-08-18T13:06:16Z","cross_cats_sorted":[],"title_canon_sha256":"a1c3ab628038d002bf24944d88fe2ccbdd8a93db55605fd1b70dd209457d9111","abstract_canon_sha256":"41b1a0d4c14e701bf5c89a46b43cdd349bec59a6b379f2298fe32bca9be00041"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:07.537311Z","signature_b64":"zTbocyRI8f5qh5B0tY3Q9S9SOCXwiUIPy2iXCnEXewW5kmi0127PjnVwmtRt8D/tRNBy+TCBb1hwQU48zXlcCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea2d69f4ec16f9b59078945d81e55628f453ce524b771d11082c9c2f85a6a5d8","last_reissued_at":"2026-05-18T04:15:07.536538Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:07.536538Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"H\\\"older continuity of Lyapunov exponent for quasi-periodic Jacobi operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Kai Tao","submitted_at":"2011-08-18T13:06:16Z","abstract_excerpt":"We consider the quasi-periodic Jacobi operator $H_{x,\\omega}$ in $l^2(\\mathbb{Z})$ $(H_{x,\\omega}\\phi)(n) = -b(x+(n+1)\\omega)\\phi(n+1) - b(x+n\\omega)\\phi(n-1) + a(x+n\\omega)\\phi(n) = E\\phi(n),\\ n\\in\\mathbb{Z},$ where $a(x),\\ b(x)$ are analytic function on $\\mathbb{T}$, $b$ is not identically zero, and $\\omega$ obeys some strong Diophantine condition.\n  We consider the corresponding unimodular cocycle. We prove that if the Lyapunov exponent $L(E)$ of the cocycle is positive for some $E=E_0$, then there exists $\\rho_0=\\rho_0(a,b,\\omega,E_0)$, $\\beta=\\beta(a,b,\\omega)$ such that $|L(E)-L(E')|<|E-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3747","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":"1108.3747","created_at":"2026-05-18T04:15:07.536687+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.3747v1","created_at":"2026-05-18T04:15:07.536687+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.3747","created_at":"2026-05-18T04:15:07.536687+00:00"},{"alias_kind":"pith_short_12","alias_value":"5IWWT5HMC343","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"5IWWT5HMC343LEDY","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"5IWWT5HM","created_at":"2026-05-18T12:26:20.644004+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/5IWWT5HMC343LEDYSROYDZKWFD","json":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD.json","graph_json":"https://pith.science/api/pith-number/5IWWT5HMC343LEDYSROYDZKWFD/graph.json","events_json":"https://pith.science/api/pith-number/5IWWT5HMC343LEDYSROYDZKWFD/events.json","paper":"https://pith.science/paper/5IWWT5HM"},"agent_actions":{"view_html":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD","download_json":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD.json","view_paper":"https://pith.science/paper/5IWWT5HM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.3747&json=true","fetch_graph":"https://pith.science/api/pith-number/5IWWT5HMC343LEDYSROYDZKWFD/graph.json","fetch_events":"https://pith.science/api/pith-number/5IWWT5HMC343LEDYSROYDZKWFD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD/action/storage_attestation","attest_author":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD/action/author_attestation","sign_citation":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD/action/citation_signature","submit_replication":"https://pith.science/pith/5IWWT5HMC343LEDYSROYDZKWFD/action/replication_record"}},"created_at":"2026-05-18T04:15:07.536687+00:00","updated_at":"2026-05-18T04:15:07.536687+00:00"}