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The shift dynamical system induced by $\\sigma_{a,b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile $\\mathcal{T}_{a,b}$ with fractal boundary.\n  We prove that $\\mathcal{T}_{a,b}$ is homeomorphic to the closed disk iff $2b-a\\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a"},"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":"1411.7544","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-11-27T10:53:04Z","cross_cats_sorted":[],"title_canon_sha256":"8a3706f6096ebed6b84696ecc23cdd8da8742a79bdaa3c733bbc08b92946d6da","abstract_canon_sha256":"7a5f9038cac4837a186d38b522fff05e9c5e03278b87ff7540ab991a017387f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:37.710680Z","signature_b64":"tBye3iKZMd/8a/Zw9fRn6Ar0+miM16X5dop1wC6/Q+hTXIZGCfQ4H/P8YmHgS298TyCWsxeQFku/Mj7rtMvrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e90f18293ea63aef0cce94b41d871fddbe744e5ba7c85df107b9f3364e11016","last_reissued_at":"2026-05-18T02:32:37.710261Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:37.710261Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Topological properties of a class of cubic Rauzy fractals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Beno\\^it Loridant","submitted_at":"2014-11-27T10:53:04Z","abstract_excerpt":"We consider the substitution $\\sigma_{a,b}$ defined by $$\\begin{array}{rlcl} \\sigma_{a,b}: & 1 & \\mapsto & \\underbrace{1\\ldots 1}_{a}2 \\\\ & 2 & \\mapsto & \\underbrace{1\\ldots 1}_{b}3 \\\\ & 3 & \\mapsto & 1 \\end{array} $$ with $a\\geq b\\geq 1$. The shift dynamical system induced by $\\sigma_{a,b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile $\\mathcal{T}_{a,b}$ with fractal boundary.\n  We prove that $\\mathcal{T}_{a,b}$ is homeomorphic to the closed disk iff $2b-a\\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.7544","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":"1411.7544","created_at":"2026-05-18T02:32:37.710324+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.7544v1","created_at":"2026-05-18T02:32:37.710324+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.7544","created_at":"2026-05-18T02:32:37.710324+00:00"},{"alias_kind":"pith_short_12","alias_value":"N2IPDAUT5JR2","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_16","alias_value":"N2IPDAUT5JR254GM","created_at":"2026-05-18T12:28:41.024544+00:00"},{"alias_kind":"pith_short_8","alias_value":"N2IPDAUT","created_at":"2026-05-18T12:28:41.024544+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/N2IPDAUT5JR254GM5FFUDWDR7X","json":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X.json","graph_json":"https://pith.science/api/pith-number/N2IPDAUT5JR254GM5FFUDWDR7X/graph.json","events_json":"https://pith.science/api/pith-number/N2IPDAUT5JR254GM5FFUDWDR7X/events.json","paper":"https://pith.science/paper/N2IPDAUT"},"agent_actions":{"view_html":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X","download_json":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X.json","view_paper":"https://pith.science/paper/N2IPDAUT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.7544&json=true","fetch_graph":"https://pith.science/api/pith-number/N2IPDAUT5JR254GM5FFUDWDR7X/graph.json","fetch_events":"https://pith.science/api/pith-number/N2IPDAUT5JR254GM5FFUDWDR7X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X/action/storage_attestation","attest_author":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X/action/author_attestation","sign_citation":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X/action/citation_signature","submit_replication":"https://pith.science/pith/N2IPDAUT5JR254GM5FFUDWDR7X/action/replication_record"}},"created_at":"2026-05-18T02:32:37.710324+00:00","updated_at":"2026-05-18T02:32:37.710324+00:00"}