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Let $P \\colon (0,\\infty)\\to \\R$ be the pressure function of $\\{M_i\\}_{i=1}^\\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property."},"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":"1009.3129","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-09-16T09:49:31Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"20cd350c3a8af56067a07f4d1661336db87fd9bdcb8d9610ab968ec79318e033","abstract_canon_sha256":"e692e17b661666d228c1a7628da19b305fb94f9fd8b1cc62857c10f00c7acbc7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:34.472381Z","signature_b64":"ELf+c6UbNo1q3JLTGbJ4B4OoQ4wnZLpuGKIfJ9tezKznuQG4TgqNIoe9G04m3xaTuJjWgyKVxZ6TcYkyhoZTDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"896d1ca0ab63ca2c498d9328fdaf86de74c014212d2fe7c3b8ad50b3f35b5d8d","last_reissued_at":"2026-05-18T00:51:34.471925Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:34.471925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equilibrium states of the pressure function for products of matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"Antti Kaenmaki, De-Jun Feng","submitted_at":"2010-09-16T09:49:31Z","abstract_excerpt":"Let $\\{M_i\\}_{i=1}^\\ell$ be a non-trivial family of $d\\times d$ complex matrices, in the sense that for any $n\\in \\N$, there exists $i_1... i_n\\in \\{1,..., \\ell\\}^n$ such that $M_{i_1}... M_{i_n}\\neq {\\bf 0}$. Let $P \\colon (0,\\infty)\\to \\R$ be the pressure function of $\\{M_i\\}_{i=1}^\\ell$. We show that for each $q>0$, there are at most $d$ ergodic $q$-equilibrium states of $P$, and each of them satisfies certain Gibbs property."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3129","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":"1009.3129","created_at":"2026-05-18T00:51:34.472001+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.3129v1","created_at":"2026-05-18T00:51:34.472001+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.3129","created_at":"2026-05-18T00:51:34.472001+00:00"},{"alias_kind":"pith_short_12","alias_value":"RFWRZIFLMPFC","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_16","alias_value":"RFWRZIFLMPFCYSMN","created_at":"2026-05-18T12:26:13.927090+00:00"},{"alias_kind":"pith_short_8","alias_value":"RFWRZIFL","created_at":"2026-05-18T12:26:13.927090+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/RFWRZIFLMPFCYSMNSMUP3L4G3Z","json":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z.json","graph_json":"https://pith.science/api/pith-number/RFWRZIFLMPFCYSMNSMUP3L4G3Z/graph.json","events_json":"https://pith.science/api/pith-number/RFWRZIFLMPFCYSMNSMUP3L4G3Z/events.json","paper":"https://pith.science/paper/RFWRZIFL"},"agent_actions":{"view_html":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z","download_json":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z.json","view_paper":"https://pith.science/paper/RFWRZIFL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.3129&json=true","fetch_graph":"https://pith.science/api/pith-number/RFWRZIFLMPFCYSMNSMUP3L4G3Z/graph.json","fetch_events":"https://pith.science/api/pith-number/RFWRZIFLMPFCYSMNSMUP3L4G3Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z/action/storage_attestation","attest_author":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z/action/author_attestation","sign_citation":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z/action/citation_signature","submit_replication":"https://pith.science/pith/RFWRZIFLMPFCYSMNSMUP3L4G3Z/action/replication_record"}},"created_at":"2026-05-18T00:51:34.472001+00:00","updated_at":"2026-05-18T00:51:34.472001+00:00"}