{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:HYTVZYKLD5XASANVWAH42KJCXC","short_pith_number":"pith:HYTVZYKL","canonical_record":{"source":{"id":"1207.2702","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-07-11T16:33:13Z","cross_cats_sorted":[],"title_canon_sha256":"e5d4504edb0733c00add5dd67ab04f12dbcdf909c6955c104314137ff96158b4","abstract_canon_sha256":"3f853a2868d1646c728951f1c33baafa6aac4618187391ef199cbc1d6a0fbb8a"},"schema_version":"1.0"},"canonical_sha256":"3e275ce14b1f6e0901b5b00fcd2922b886c00b196b822cfacbb7de939e5c630b","source":{"kind":"arxiv","id":"1207.2702","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2702","created_at":"2026-05-18T03:51:14Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2702v1","created_at":"2026-05-18T03:51:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2702","created_at":"2026-05-18T03:51:14Z"},{"alias_kind":"pith_short_12","alias_value":"HYTVZYKLD5XA","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HYTVZYKLD5XASANV","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HYTVZYKL","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:HYTVZYKLD5XASANVWAH42KJCXC","target":"record","payload":{"canonical_record":{"source":{"id":"1207.2702","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-07-11T16:33:13Z","cross_cats_sorted":[],"title_canon_sha256":"e5d4504edb0733c00add5dd67ab04f12dbcdf909c6955c104314137ff96158b4","abstract_canon_sha256":"3f853a2868d1646c728951f1c33baafa6aac4618187391ef199cbc1d6a0fbb8a"},"schema_version":"1.0"},"canonical_sha256":"3e275ce14b1f6e0901b5b00fcd2922b886c00b196b822cfacbb7de939e5c630b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:14.753782Z","signature_b64":"u7CQ4LrVggPpHxMImve97J2y3RL6eH8OqgmxA549EKI/otlpGx5+43IGI85qaqrWu8y3j/u+lsQUdquY5nTuBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3e275ce14b1f6e0901b5b00fcd2922b886c00b196b822cfacbb7de939e5c630b","last_reissued_at":"2026-05-18T03:51:14.752984Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:14.752984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.2702","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1FD+c13Wog8JF4eXHK8l/38f/lZ8GIAOM1bFhbSg2xAGBQYb6zj5tTBYoo7klX1ZSWYKkJptl17mcPbtU4AzAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T02:47:37.850413Z"},"content_sha256":"1b9281dfbcaed4c4455c918b5e386bfc5e552ba21c7453e7cfaccc9a0ec7f4cb","schema_version":"1.0","event_id":"sha256:1b9281dfbcaed4c4455c918b5e386bfc5e552ba21c7453e7cfaccc9a0ec7f4cb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:HYTVZYKLD5XASANVWAH42KJCXC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Rui Gao, Weixiao Shen","submitted_at":"2012-07-11T16:33:13Z","abstract_excerpt":"We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2702","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1de4ft/pmUznlYJv0j7wJnL4Mw1TNPDxaNA/0hJhAcx32Ho45HdOR9KZ/3bm3RZBaDzotsSbZLoYbHuklxLXAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T02:47:37.851073Z"},"content_sha256":"4d8bc0966755bcfff39ad7a46004eb2c9f0051017e96fbe0efe012f59b5033e6","schema_version":"1.0","event_id":"sha256:4d8bc0966755bcfff39ad7a46004eb2c9f0051017e96fbe0efe012f59b5033e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HYTVZYKLD5XASANVWAH42KJCXC/bundle.json","state_url":"https://pith.science/pith/HYTVZYKLD5XASANVWAH42KJCXC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HYTVZYKLD5XASANVWAH42KJCXC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T02:47:37Z","links":{"resolver":"https://pith.science/pith/HYTVZYKLD5XASANVWAH42KJCXC","bundle":"https://pith.science/pith/HYTVZYKLD5XASANVWAH42KJCXC/bundle.json","state":"https://pith.science/pith/HYTVZYKLD5XASANVWAH42KJCXC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HYTVZYKLD5XASANVWAH42KJCXC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:HYTVZYKLD5XASANVWAH42KJCXC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"3f853a2868d1646c728951f1c33baafa6aac4618187391ef199cbc1d6a0fbb8a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-07-11T16:33:13Z","title_canon_sha256":"e5d4504edb0733c00add5dd67ab04f12dbcdf909c6955c104314137ff96158b4"},"schema_version":"1.0","source":{"id":"1207.2702","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2702","created_at":"2026-05-18T03:51:14Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2702v1","created_at":"2026-05-18T03:51:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2702","created_at":"2026-05-18T03:51:14Z"},{"alias_kind":"pith_short_12","alias_value":"HYTVZYKLD5XA","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"HYTVZYKLD5XASANV","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"HYTVZYKL","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:4d8bc0966755bcfff39ad7a46004eb2c9f0051017e96fbe0efe012f59b5033e6","target":"graph","created_at":"2026-05-18T03:51:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider skew-products of quadratic maps over certain Misiurewicz-Thurston maps and study their statistical properties. We prove that, when the coupling function is a polynomial of odd degree, such a system admits two positive Lyapunov exponents almost everywhere and a unique absolutely continuous invariant probability measure.","authors_text":"Rui Gao, Weixiao Shen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-07-11T16:33:13Z","title":"Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2702","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1b9281dfbcaed4c4455c918b5e386bfc5e552ba21c7453e7cfaccc9a0ec7f4cb","target":"record","created_at":"2026-05-18T03:51:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"3f853a2868d1646c728951f1c33baafa6aac4618187391ef199cbc1d6a0fbb8a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-07-11T16:33:13Z","title_canon_sha256":"e5d4504edb0733c00add5dd67ab04f12dbcdf909c6955c104314137ff96158b4"},"schema_version":"1.0","source":{"id":"1207.2702","kind":"arxiv","version":1}},"canonical_sha256":"3e275ce14b1f6e0901b5b00fcd2922b886c00b196b822cfacbb7de939e5c630b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3e275ce14b1f6e0901b5b00fcd2922b886c00b196b822cfacbb7de939e5c630b","first_computed_at":"2026-05-18T03:51:14.752984Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:14.752984Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"u7CQ4LrVggPpHxMImve97J2y3RL6eH8OqgmxA549EKI/otlpGx5+43IGI85qaqrWu8y3j/u+lsQUdquY5nTuBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:14.753782Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2702","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1b9281dfbcaed4c4455c918b5e386bfc5e552ba21c7453e7cfaccc9a0ec7f4cb","sha256:4d8bc0966755bcfff39ad7a46004eb2c9f0051017e96fbe0efe012f59b5033e6"],"state_sha256":"fadd2ddded63aef5ea6eaf7905460dce63ea855375a8fbad4ec296e2e25e27d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aat7jD3W5Gdc0kvbwjF2DVGuUhJSOlh8D8y+NlsjAodmtXA/D+6dgkVPnFjfo8d+5Eoo5FL1KR3suvHpGU4dCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T02:47:37.854595Z","bundle_sha256":"4b273922b83a9e539e93dba413ef808fd6df8f74025c05d67aec6316dc2854db"}}