{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:T4Y2CSGKN4JPMDCIKBEHCY5XQN","short_pith_number":"pith:T4Y2CSGK","canonical_record":{"source":{"id":"1512.03171","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-10T08:36:07Z","cross_cats_sorted":[],"title_canon_sha256":"9402dc34ab5f0a966ebd3242cd2ab2b1e16b08afc69b3d5df471af405e5ec354","abstract_canon_sha256":"a3f80df28e17c7aca67e0ec50f3a59f3ca3434ef93fbe9c7dde9ca687f630938"},"schema_version":"1.0"},"canonical_sha256":"9f31a148ca6f12f60c4850487163b78372b3c460d5f522bf2226fa72cba938ac","source":{"kind":"arxiv","id":"1512.03171","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03171","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03171v2","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03171","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"T4Y2CSGKN4JP","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"T4Y2CSGKN4JPMDCI","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"T4Y2CSGK","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:T4Y2CSGKN4JPMDCIKBEHCY5XQN","target":"record","payload":{"canonical_record":{"source":{"id":"1512.03171","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-10T08:36:07Z","cross_cats_sorted":[],"title_canon_sha256":"9402dc34ab5f0a966ebd3242cd2ab2b1e16b08afc69b3d5df471af405e5ec354","abstract_canon_sha256":"a3f80df28e17c7aca67e0ec50f3a59f3ca3434ef93fbe9c7dde9ca687f630938"},"schema_version":"1.0"},"canonical_sha256":"9f31a148ca6f12f60c4850487163b78372b3c460d5f522bf2226fa72cba938ac","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:54.551793Z","signature_b64":"hnH1QcAODc8eAMWtnlg819RSU9XF/s01j18dKC5a5zcOL7zF0IY472vZj714OUfz9PVZsMessg0lpTEww72wDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f31a148ca6f12f60c4850487163b78372b3c460d5f522bf2226fa72cba938ac","last_reissued_at":"2026-05-18T00:24:54.551096Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:54.551096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.03171","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0c3t3Dupcrg/ggQrfiugF6oWVlzM4fzv6J091wOEp7hdZNxXYYCYzRGWk6Uh3FBZdiDrh3xwwKwo0p5dbRQkCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:20:51.059801Z"},"content_sha256":"c04c8021d7bffd4383a8f2665deb703b3c25d0af4c4faa8a6115f34ce711500c","schema_version":"1.0","event_id":"sha256:c04c8021d7bffd4383a8f2665deb703b3c25d0af4c4faa8a6115f34ce711500c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:T4Y2CSGKN4JPMDCIKBEHCY5XQN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Indestructible dynamics of torus maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"James Yorke, Suddhasattwa Das","submitted_at":"2015-12-10T08:36:07Z","abstract_excerpt":"Given a $d$-dimensional torus map $F(z)=Mz+G(z)\\bmod 1$, where $M$ is an integer-matrix and and $G$ is a periodic function, we find conditions on $M$ under which $F$ is semi-conjugate to a linear torus map, independently of $G$. We also find a conditions $G$ under which these semi-conjugacies can be turned into conjugacies. These conditions are satisfied by open sets of torus maps (in the $C^1$-topology) and therefore describe some asymptotic behavior of trajectories which are stable under perturbations to the map."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03171","kind":"arxiv","version":2},"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":"2026-05-18T00:24:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r1CEBAjS0ZFoW/WJtER9teXTIzlC/KygU/19f9XHgJznJM2UA6J/kyBxz6ijwiS2IvWqUQc2AcqcPr1sW9bWAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T23:20:51.060529Z"},"content_sha256":"fcfa352df584ef301b44aa60a28b1cc39160cf1f680ec5d72e6e4810221e495d","schema_version":"1.0","event_id":"sha256:fcfa352df584ef301b44aa60a28b1cc39160cf1f680ec5d72e6e4810221e495d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN/bundle.json","state_url":"https://pith.science/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN/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":"2026-06-22T23:20:51Z","links":{"resolver":"https://pith.science/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN","bundle":"https://pith.science/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN/bundle.json","state":"https://pith.science/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T4Y2CSGKN4JPMDCIKBEHCY5XQN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:T4Y2CSGKN4JPMDCIKBEHCY5XQN","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":"a3f80df28e17c7aca67e0ec50f3a59f3ca3434ef93fbe9c7dde9ca687f630938","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-10T08:36:07Z","title_canon_sha256":"9402dc34ab5f0a966ebd3242cd2ab2b1e16b08afc69b3d5df471af405e5ec354"},"schema_version":"1.0","source":{"id":"1512.03171","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.03171","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"arxiv_version","alias_value":"1512.03171v2","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.03171","created_at":"2026-05-18T00:24:54Z"},{"alias_kind":"pith_short_12","alias_value":"T4Y2CSGKN4JP","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"T4Y2CSGKN4JPMDCI","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"T4Y2CSGK","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:fcfa352df584ef301b44aa60a28b1cc39160cf1f680ec5d72e6e4810221e495d","target":"graph","created_at":"2026-05-18T00:24:54Z","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":"Given a $d$-dimensional torus map $F(z)=Mz+G(z)\\bmod 1$, where $M$ is an integer-matrix and and $G$ is a periodic function, we find conditions on $M$ under which $F$ is semi-conjugate to a linear torus map, independently of $G$. We also find a conditions $G$ under which these semi-conjugacies can be turned into conjugacies. These conditions are satisfied by open sets of torus maps (in the $C^1$-topology) and therefore describe some asymptotic behavior of trajectories which are stable under perturbations to the map.","authors_text":"James Yorke, Suddhasattwa Das","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-10T08:36:07Z","title":"Indestructible dynamics of torus maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.03171","kind":"arxiv","version":2},"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:c04c8021d7bffd4383a8f2665deb703b3c25d0af4c4faa8a6115f34ce711500c","target":"record","created_at":"2026-05-18T00:24:54Z","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":"a3f80df28e17c7aca67e0ec50f3a59f3ca3434ef93fbe9c7dde9ca687f630938","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-12-10T08:36:07Z","title_canon_sha256":"9402dc34ab5f0a966ebd3242cd2ab2b1e16b08afc69b3d5df471af405e5ec354"},"schema_version":"1.0","source":{"id":"1512.03171","kind":"arxiv","version":2}},"canonical_sha256":"9f31a148ca6f12f60c4850487163b78372b3c460d5f522bf2226fa72cba938ac","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9f31a148ca6f12f60c4850487163b78372b3c460d5f522bf2226fa72cba938ac","first_computed_at":"2026-05-18T00:24:54.551096Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:54.551096Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hnH1QcAODc8eAMWtnlg819RSU9XF/s01j18dKC5a5zcOL7zF0IY472vZj714OUfz9PVZsMessg0lpTEww72wDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:54.551793Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.03171","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c04c8021d7bffd4383a8f2665deb703b3c25d0af4c4faa8a6115f34ce711500c","sha256:fcfa352df584ef301b44aa60a28b1cc39160cf1f680ec5d72e6e4810221e495d"],"state_sha256":"0299bae9db4c17bd8b29641395a758903f751c7d19fc42af46e52abc4fc9be5c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RZQoi8iDAjh2uuaGZR/9TUkStZq+4Xo6yXn75/tkhJQHEPCNP+coO22efBFeCBiwnX3stxIBKrsJyW3zvgxPAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T23:20:51.062447Z","bundle_sha256":"04823285d5d05df2afb688cf7c4d845d3f3a4bebf37eba86697e6cb3c64ba2a4"}}