{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:NZTAWZI54XNXB3PIWBX25HEXHW","short_pith_number":"pith:NZTAWZI5","schema_version":"1.0","canonical_sha256":"6e660b651de5db70ede8b06fae9c973d8457c89385bb16d3b86edcb87347b876","source":{"kind":"arxiv","id":"1307.1090","version":1},"attestation_state":"computed","paper":{"title":"Countable Contraction Maps in Metric Spaces: Invariant Sets and Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Maria Fernanda Barrozo, Ursula Molter","submitted_at":"2013-07-03T17:46:13Z","abstract_excerpt":"We consider a complete metric space $(X,d)$ and a countable number of contractive mappings on $X$, $\\mathcal{F}=\\{F_i:i\\in\\mathbb N\\}$. We show the existence of a {\\em smallest} invariant set (with respect to inclusion) for $\\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\\x) = r_i \\boldmath{x} + b_i$ on $X=\\mathbb{R}^d$, we can prove a converse of the classic result on contraction maps. Precisely, we can show that for that case, there exists a {\\em unique} bounded invariant set if and only if $r = \\sup_i r_i$ is strictly smaller than 1.\n  Further, if $\\rho = \\{\\rho_k\\}_{k\\in \\mathbb N}$ "},"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":"1307.1090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-07-03T17:46:13Z","cross_cats_sorted":[],"title_canon_sha256":"9363c62fd55035e8d29b790505d4cfa4146cede4b83b2602aed62725567b9d32","abstract_canon_sha256":"eab312655baed3fb0fae3e550bb6b400fbaae70a3a17b31478bdacf4117b4144"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:22.899306Z","signature_b64":"49YcCWc7z/gkvzz+GrcF70vkRn+m06HFL+ozwgSTX/RhW/ARkMRpjmdkHeX7WBw1TAbj5CZpz1RKqlk7qvigBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e660b651de5db70ede8b06fae9c973d8457c89385bb16d3b86edcb87347b876","last_reissued_at":"2026-05-18T03:19:22.898700Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:22.898700Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Countable Contraction Maps in Metric Spaces: Invariant Sets and Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Maria Fernanda Barrozo, Ursula Molter","submitted_at":"2013-07-03T17:46:13Z","abstract_excerpt":"We consider a complete metric space $(X,d)$ and a countable number of contractive mappings on $X$, $\\mathcal{F}=\\{F_i:i\\in\\mathbb N\\}$. We show the existence of a {\\em smallest} invariant set (with respect to inclusion) for $\\mathcal{F}$. If the maps $F_i$ are of the form $F_i(\\x) = r_i \\boldmath{x} + b_i$ on $X=\\mathbb{R}^d$, we can prove a converse of the classic result on contraction maps. Precisely, we can show that for that case, there exists a {\\em unique} bounded invariant set if and only if $r = \\sup_i r_i$ is strictly smaller than 1.\n  Further, if $\\rho = \\{\\rho_k\\}_{k\\in \\mathbb N}$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1090","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":"1307.1090","created_at":"2026-05-18T03:19:22.898791+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.1090v1","created_at":"2026-05-18T03:19:22.898791+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1090","created_at":"2026-05-18T03:19:22.898791+00:00"},{"alias_kind":"pith_short_12","alias_value":"NZTAWZI54XNX","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_16","alias_value":"NZTAWZI54XNXB3PI","created_at":"2026-05-18T12:27:54.935989+00:00"},{"alias_kind":"pith_short_8","alias_value":"NZTAWZI5","created_at":"2026-05-18T12:27:54.935989+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/NZTAWZI54XNXB3PIWBX25HEXHW","json":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW.json","graph_json":"https://pith.science/api/pith-number/NZTAWZI54XNXB3PIWBX25HEXHW/graph.json","events_json":"https://pith.science/api/pith-number/NZTAWZI54XNXB3PIWBX25HEXHW/events.json","paper":"https://pith.science/paper/NZTAWZI5"},"agent_actions":{"view_html":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW","download_json":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW.json","view_paper":"https://pith.science/paper/NZTAWZI5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.1090&json=true","fetch_graph":"https://pith.science/api/pith-number/NZTAWZI54XNXB3PIWBX25HEXHW/graph.json","fetch_events":"https://pith.science/api/pith-number/NZTAWZI54XNXB3PIWBX25HEXHW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW/action/storage_attestation","attest_author":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW/action/author_attestation","sign_citation":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW/action/citation_signature","submit_replication":"https://pith.science/pith/NZTAWZI54XNXB3PIWBX25HEXHW/action/replication_record"}},"created_at":"2026-05-18T03:19:22.898791+00:00","updated_at":"2026-05-18T03:19:22.898791+00:00"}