{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4OMUYZYC6QW5OEGYHS7MMTE357","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":"fa95a958ee1abbf958563c1fd50cde934020922db9336799ea2c31b1e0112b6f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-25T20:06:55Z","title_canon_sha256":"61c976de4f31b5575459219b2e779ee355f154dc15d9dbc5ef6caa3a8d7ef558"},"schema_version":"1.0","source":{"id":"1204.5755","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.5755","created_at":"2026-05-18T03:56:59Z"},{"alias_kind":"arxiv_version","alias_value":"1204.5755v1","created_at":"2026-05-18T03:56:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.5755","created_at":"2026-05-18T03:56:59Z"},{"alias_kind":"pith_short_12","alias_value":"4OMUYZYC6QW5","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4OMUYZYC6QW5OEGY","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4OMUYZYC","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:7b693f35a2ac12ba66d7cbd1e625c84970c8f801320a936d97de7ce1812a9bfa","target":"graph","created_at":"2026-05-18T03:56:59Z","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":"Return words constitute a powerful tool for studying symbolic dynamical systems. They may be regarded as a discrete analogue of the first return map in dynamical systems. In this paper we investigate two abelian variants of the notion of return word, each of them gives rise to a new characterization of Sturmian words. We prove that a recurrent infinite word is Sturmian if and only if each of its factors has two or three abelian (or semi-abelian) returns. We study the structure of abelian returns in Sturmian words and give a characterization of those factors having exactly two abelian returns. ","authors_text":"Luca Q. Zamboni, Svetlana Puzynina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-25T20:06:55Z","title":"Abelian returns in Sturmian words"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.5755","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":"sha256:271ce55379375c9fc0eda8c9172667904ebf9a357d9e53699ca269cf4c83141c","target":"record","created_at":"2026-05-18T03:56:59Z","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":"fa95a958ee1abbf958563c1fd50cde934020922db9336799ea2c31b1e0112b6f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-25T20:06:55Z","title_canon_sha256":"61c976de4f31b5575459219b2e779ee355f154dc15d9dbc5ef6caa3a8d7ef558"},"schema_version":"1.0","source":{"id":"1204.5755","kind":"arxiv","version":1}},"canonical_sha256":"e3994c6702f42dd710d83cbec64c9beffe534f31e5ca10c64e15c0230e03366b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e3994c6702f42dd710d83cbec64c9beffe534f31e5ca10c64e15c0230e03366b","first_computed_at":"2026-05-18T03:56:59.709222Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:56:59.709222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LGiXtHEIyPUD9l18fxPVS0vEnto5Ae3iMnTDjU6SNRDlHu2X5RQJiViMzhu5C5sL5cqrYFcG9vXuZn+FeZWgBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:56:59.710000Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.5755","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:271ce55379375c9fc0eda8c9172667904ebf9a357d9e53699ca269cf4c83141c","sha256:7b693f35a2ac12ba66d7cbd1e625c84970c8f801320a936d97de7ce1812a9bfa"],"state_sha256":"808aac5f359d3592bc9c5d4c0efb7621f900eb9875e072bc56fb55ce36e7e886"}