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They described the set-valued bifurcation set\n  \\[\n  \\mathcal E_\\beta:=\\{t\\in[0, 1): K_\\beta(t')\\ne K_\\beta(t)~\\forall t'>t\\},\n  \\]\n  where $K_\\beta(t):=\\{x\\in[0, 1): T_\\beta^n(x)\\ge t~\\forall n\\ge 0\\}$ is the survivor set. In this paper we investigate the dimension bifurcation set\n  \\[\n  \\mathcal B_\\beta:=\\{t\\in[0, 1): \\dim_H K_\\beta(t')\\ne \\dim_H K_\\beta(t)~\\forall t'>t\\},\n  \\]\n  where $\\dim_H$ denotes the Hausdorff dimensio"},"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":"1904.07007","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-04-15T12:50:57Z","cross_cats_sorted":[],"title_canon_sha256":"c957992fe479c694f8bbe902ad8d1994d15f8d2f4804c7b6fb87fdf6e020949a","abstract_canon_sha256":"5fb7823503e0b057c1657c643cb1555aa8f14f6bd309d9a3c0f070d5d02c07c8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:35.491032Z","signature_b64":"WacOrCwYs3ykswxl7Bp4FKEFMGAcHKnTCsHeOWhaFbFgkDSK7vNrfTnh1NPpvC7GSjBXYkTiGUOuBKSZ9ZFGCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d4f438a411dfd9930b82cabbdaf06c5581ee53d3a6b53745d46b03b353777db","last_reissued_at":"2026-05-17T23:48:35.490556Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:35.490556Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two bifurcation sets arising from the beta transformation with a hole at $0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Derong Kong, Simon Baker","submitted_at":"2019-04-15T12:50:57Z","abstract_excerpt":"Given $\\beta\\in(1,2],$ the $\\beta$-transformation $T_\\beta: x\\mapsto \\beta x\\pmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-valued bifurcation set\n  \\[\n  \\mathcal E_\\beta:=\\{t\\in[0, 1): K_\\beta(t')\\ne K_\\beta(t)~\\forall t'>t\\},\n  \\]\n  where $K_\\beta(t):=\\{x\\in[0, 1): T_\\beta^n(x)\\ge t~\\forall n\\ge 0\\}$ is the survivor set. In this paper we investigate the dimension bifurcation set\n  \\[\n  \\mathcal B_\\beta:=\\{t\\in[0, 1): \\dim_H K_\\beta(t')\\ne \\dim_H K_\\beta(t)~\\forall t'>t\\},\n  \\]\n  where $\\dim_H$ denotes the Hausdorff dimensio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.07007","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":"1904.07007","created_at":"2026-05-17T23:48:35.490617+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.07007v1","created_at":"2026-05-17T23:48:35.490617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.07007","created_at":"2026-05-17T23:48:35.490617+00:00"},{"alias_kind":"pith_short_12","alias_value":"BVHUHCSBDX6Z","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_16","alias_value":"BVHUHCSBDX6ZSMFY","created_at":"2026-05-18T12:33:12.712433+00:00"},{"alias_kind":"pith_short_8","alias_value":"BVHUHCSB","created_at":"2026-05-18T12:33:12.712433+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/BVHUHCSBDX6ZSMFYFSV33LYGYV","json":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV.json","graph_json":"https://pith.science/api/pith-number/BVHUHCSBDX6ZSMFYFSV33LYGYV/graph.json","events_json":"https://pith.science/api/pith-number/BVHUHCSBDX6ZSMFYFSV33LYGYV/events.json","paper":"https://pith.science/paper/BVHUHCSB"},"agent_actions":{"view_html":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV","download_json":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV.json","view_paper":"https://pith.science/paper/BVHUHCSB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.07007&json=true","fetch_graph":"https://pith.science/api/pith-number/BVHUHCSBDX6ZSMFYFSV33LYGYV/graph.json","fetch_events":"https://pith.science/api/pith-number/BVHUHCSBDX6ZSMFYFSV33LYGYV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV/action/storage_attestation","attest_author":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV/action/author_attestation","sign_citation":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV/action/citation_signature","submit_replication":"https://pith.science/pith/BVHUHCSBDX6ZSMFYFSV33LYGYV/action/replication_record"}},"created_at":"2026-05-17T23:48:35.490617+00:00","updated_at":"2026-05-17T23:48:35.490617+00:00"}