{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:NM7XQLEXPCF3LTJXNE5QGAKCGY","short_pith_number":"pith:NM7XQLEX","schema_version":"1.0","canonical_sha256":"6b3f782c97788bb5cd37693b03014236292dd74e7497c3959df6ec636f24c9a1","source":{"kind":"arxiv","id":"1001.5323","version":2},"attestation_state":"computed","paper":{"title":"Class Degree and Relative Maximal Entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anthony Quas, Mahsa Allahbakhshi","submitted_at":"2010-01-29T04:56:44Z","abstract_excerpt":"Given a factor code $\\pi$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $\\pi$ is finite-to-one there is an invariant called the degree of $\\pi$ which is defined the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $\\nu$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $\\nu$ and have maximal entropy among all measures in the fibre $\\pi^{-1}\\{\\nu"},"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":"1001.5323","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-01-29T04:56:44Z","cross_cats_sorted":[],"title_canon_sha256":"50129d1162afaae0fa07103df02d4b5ec9e75aaaabbd89357a4d17a5b20f5651","abstract_canon_sha256":"1891e655a8683a764347970027cb6ce9ed5b3a27add16d5a52991b529fe28e89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:22.854990Z","signature_b64":"fplgKOJgZzigYMWCpt0JIQ/T8Gh4OAHSIvaFfINsTu8MtBLkZsNc3htDsg2Jf3aoT+vAYTC+XvoZCyveJnK6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b3f782c97788bb5cd37693b03014236292dd74e7497c3959df6ec636f24c9a1","last_reissued_at":"2026-05-18T03:06:22.854145Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:22.854145Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Class Degree and Relative Maximal Entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anthony Quas, Mahsa Allahbakhshi","submitted_at":"2010-01-29T04:56:44Z","abstract_excerpt":"Given a factor code $\\pi$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $\\pi$ is finite-to-one there is an invariant called the degree of $\\pi$ which is defined the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $\\nu$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $\\nu$ and have maximal entropy among all measures in the fibre $\\pi^{-1}\\{\\nu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.5323","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1001.5323","created_at":"2026-05-18T03:06:22.854306+00:00"},{"alias_kind":"arxiv_version","alias_value":"1001.5323v2","created_at":"2026-05-18T03:06:22.854306+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.5323","created_at":"2026-05-18T03:06:22.854306+00:00"},{"alias_kind":"pith_short_12","alias_value":"NM7XQLEXPCF3","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"NM7XQLEXPCF3LTJX","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"NM7XQLEX","created_at":"2026-05-18T12:26:10.704358+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/NM7XQLEXPCF3LTJXNE5QGAKCGY","json":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY.json","graph_json":"https://pith.science/api/pith-number/NM7XQLEXPCF3LTJXNE5QGAKCGY/graph.json","events_json":"https://pith.science/api/pith-number/NM7XQLEXPCF3LTJXNE5QGAKCGY/events.json","paper":"https://pith.science/paper/NM7XQLEX"},"agent_actions":{"view_html":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY","download_json":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY.json","view_paper":"https://pith.science/paper/NM7XQLEX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1001.5323&json=true","fetch_graph":"https://pith.science/api/pith-number/NM7XQLEXPCF3LTJXNE5QGAKCGY/graph.json","fetch_events":"https://pith.science/api/pith-number/NM7XQLEXPCF3LTJXNE5QGAKCGY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY/action/storage_attestation","attest_author":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY/action/author_attestation","sign_citation":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY/action/citation_signature","submit_replication":"https://pith.science/pith/NM7XQLEXPCF3LTJXNE5QGAKCGY/action/replication_record"}},"created_at":"2026-05-18T03:06:22.854306+00:00","updated_at":"2026-05-18T03:06:22.854306+00:00"}