{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:YXWNOS3JUD6BAFRWXDSWQG6DTY","short_pith_number":"pith:YXWNOS3J","schema_version":"1.0","canonical_sha256":"c5ecd74b69a0fc101636b8e5681bc39e040f8afb907fe5ae51f169ac989a38ae","source":{"kind":"arxiv","id":"1511.06893","version":1},"attestation_state":"computed","paper":{"title":"On self-affine measures with equal Hausdorff and Lyapunov dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ariel Rapaport","submitted_at":"2015-11-21T15:52:12Z","abstract_excerpt":"Let $\\mu$ be a self-affine measure on $\\mathbb{R}^{d}$ associated to a self-affine IFS $\\{\\varphi_{\\lambda}(x) = A_{\\lambda}x + v_{\\lambda}\\}_{\\lambda\\in\\Lambda}$ and a probability vector $p=(p_{\\lambda})_{\\lambda}>0$. Assume the strong separation condition holds. Let $\\gamma_{1}\\ge...\\ge\\gamma_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\\{A_{\\lambda}\\}_{\\lambda\\in\\Lambda}$ and $p^{\\mathbb{N}}$, and let $\\mathbf{G}$ be the group generated by $\\{A_{\\lambda}\\}_{\\lambda\\in\\Lambda}$. We show that if $\\gamma_{m+1}>\\gamma_{m}=...=\\gamma_{d}$, if $\\mathbf{G}$ acts irreducib"},"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":"1511.06893","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-11-21T15:52:12Z","cross_cats_sorted":[],"title_canon_sha256":"27b46703e972fcf3c5cd10dd3e87f1d8f8e8820381467bdcc8c86857c6140fc3","abstract_canon_sha256":"4d1263bfcdcc52711c778910fa2de9aec425282abc2bf0b966449f8bab97f98b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:17.206993Z","signature_b64":"+19tTw3hvQHlE2znnj700FtRRvnm0DOGlfIKDiC0lAX2IvLP9Dc5ozWB4rAH4jFjC9qVzh0rc6w/RSC4UiOYBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5ecd74b69a0fc101636b8e5681bc39e040f8afb907fe5ae51f169ac989a38ae","last_reissued_at":"2026-05-18T01:26:17.206381Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:17.206381Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On self-affine measures with equal Hausdorff and Lyapunov dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ariel Rapaport","submitted_at":"2015-11-21T15:52:12Z","abstract_excerpt":"Let $\\mu$ be a self-affine measure on $\\mathbb{R}^{d}$ associated to a self-affine IFS $\\{\\varphi_{\\lambda}(x) = A_{\\lambda}x + v_{\\lambda}\\}_{\\lambda\\in\\Lambda}$ and a probability vector $p=(p_{\\lambda})_{\\lambda}>0$. Assume the strong separation condition holds. Let $\\gamma_{1}\\ge...\\ge\\gamma_{d}$ and $D$ be the Lyapunov exponents and dimension corresponding to $\\{A_{\\lambda}\\}_{\\lambda\\in\\Lambda}$ and $p^{\\mathbb{N}}$, and let $\\mathbf{G}$ be the group generated by $\\{A_{\\lambda}\\}_{\\lambda\\in\\Lambda}$. We show that if $\\gamma_{m+1}>\\gamma_{m}=...=\\gamma_{d}$, if $\\mathbf{G}$ acts irreducib"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.06893","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":"1511.06893","created_at":"2026-05-18T01:26:17.206481+00:00"},{"alias_kind":"arxiv_version","alias_value":"1511.06893v1","created_at":"2026-05-18T01:26:17.206481+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.06893","created_at":"2026-05-18T01:26:17.206481+00:00"},{"alias_kind":"pith_short_12","alias_value":"YXWNOS3JUD6B","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_16","alias_value":"YXWNOS3JUD6BAFRW","created_at":"2026-05-18T12:29:52.810259+00:00"},{"alias_kind":"pith_short_8","alias_value":"YXWNOS3J","created_at":"2026-05-18T12:29:52.810259+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/YXWNOS3JUD6BAFRWXDSWQG6DTY","json":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY.json","graph_json":"https://pith.science/api/pith-number/YXWNOS3JUD6BAFRWXDSWQG6DTY/graph.json","events_json":"https://pith.science/api/pith-number/YXWNOS3JUD6BAFRWXDSWQG6DTY/events.json","paper":"https://pith.science/paper/YXWNOS3J"},"agent_actions":{"view_html":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY","download_json":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY.json","view_paper":"https://pith.science/paper/YXWNOS3J","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1511.06893&json=true","fetch_graph":"https://pith.science/api/pith-number/YXWNOS3JUD6BAFRWXDSWQG6DTY/graph.json","fetch_events":"https://pith.science/api/pith-number/YXWNOS3JUD6BAFRWXDSWQG6DTY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY/action/storage_attestation","attest_author":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY/action/author_attestation","sign_citation":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY/action/citation_signature","submit_replication":"https://pith.science/pith/YXWNOS3JUD6BAFRWXDSWQG6DTY/action/replication_record"}},"created_at":"2026-05-18T01:26:17.206481+00:00","updated_at":"2026-05-18T01:26:17.206481+00:00"}