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A subset $S$ of the set of integers ${\\mathbb Z}$ is said to be realizable if there is a topological space $X$ and a bijection $f:X\\to X$ such that $S={\\mathcal C}(X,f)$. A subset $S$ of ${\\mathbb Z}$ containing 0 is called a submonoid of ${\\mathbb Z}$ if the sum of any two elements of $S$ is also an element of $S$. We show that a subset $S$ of ${\\mathbb Z}$ is realizable if"},"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":"1310.1804","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2013-10-07T14:48:34Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"627de4bc09277fe7a1bdee58b20fe21e673ff54d8dfc997044290b4c6fd930b2","abstract_canon_sha256":"6564ca0a751f1b188153832e08714913cefbe3a3459ee174fd15ec970d6e6634"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:35.927694Z","signature_b64":"Wf0Idb+cLG81vkNgvsdb83cNDdQqnIVqg8sqiqItv7u+LK0P/BA8FINFujZjk/8KCo216CxS4nwrVDVYGSqIBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9494d24a9bcd5dd07d1eda0390024b16eaf8f801c51e0b47aa1d6093e6d0580e","last_reissued_at":"2026-05-18T03:08:35.927256Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:35.927256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discontinuous maps whose iterations are continuous","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.GT","authors_text":"Kouki Taniyama","submitted_at":"2013-10-07T14:48:34Z","abstract_excerpt":"Let $X$ be a topological space and $f:X\\to X$ a bijection. Let ${\\mathcal C}(X,f)$ be a set of integers such that an integer $n$ is an element of ${\\mathcal C}(X,f)$ if and only if the bijection $f^n:X\\to X$ is continuous. A subset $S$ of the set of integers ${\\mathbb Z}$ is said to be realizable if there is a topological space $X$ and a bijection $f:X\\to X$ such that $S={\\mathcal C}(X,f)$. A subset $S$ of ${\\mathbb Z}$ containing 0 is called a submonoid of ${\\mathbb Z}$ if the sum of any two elements of $S$ is also an element of $S$. We show that a subset $S$ of ${\\mathbb Z}$ is realizable if"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1804","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":"1310.1804","created_at":"2026-05-18T03:08:35.927321+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.1804v2","created_at":"2026-05-18T03:08:35.927321+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1804","created_at":"2026-05-18T03:08:35.927321+00:00"},{"alias_kind":"pith_short_12","alias_value":"SSKNESU3ZVO5","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"SSKNESU3ZVO5A7I6","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"SSKNESU3","created_at":"2026-05-18T12:27:59.945178+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/SSKNESU3ZVO5A7I63IBZAASLC3","json":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3.json","graph_json":"https://pith.science/api/pith-number/SSKNESU3ZVO5A7I63IBZAASLC3/graph.json","events_json":"https://pith.science/api/pith-number/SSKNESU3ZVO5A7I63IBZAASLC3/events.json","paper":"https://pith.science/paper/SSKNESU3"},"agent_actions":{"view_html":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3","download_json":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3.json","view_paper":"https://pith.science/paper/SSKNESU3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.1804&json=true","fetch_graph":"https://pith.science/api/pith-number/SSKNESU3ZVO5A7I63IBZAASLC3/graph.json","fetch_events":"https://pith.science/api/pith-number/SSKNESU3ZVO5A7I63IBZAASLC3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3/action/storage_attestation","attest_author":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3/action/author_attestation","sign_citation":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3/action/citation_signature","submit_replication":"https://pith.science/pith/SSKNESU3ZVO5A7I63IBZAASLC3/action/replication_record"}},"created_at":"2026-05-18T03:08:35.927321+00:00","updated_at":"2026-05-18T03:08:35.927321+00:00"}