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Let $\\mathcal{F}_t$ (resp. $\\mathcal{F}_{ip}$, $\\mathcal{F}_{fip}$) be consisting of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets).\n  The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either strongly $\\mathcal{F}_{fip"},"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":"1605.01119","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-05-04T00:17:20Z","cross_cats_sorted":[],"title_canon_sha256":"a334d585258cd1b433fe13205539e7ab4ca0c9c9e6aa189206ad299f2ec2d0f9","abstract_canon_sha256":"387818505291abc3b92f506565a616182a691319c73d189f00034600c483a400"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:58.449250Z","signature_b64":"X7UHooRxYwTdy/My8pK7w3n36MRM59wb+L1oFJlwQnFx0Xjt3kMxv8rEJxlhqYuxW1/7+8U8qn0nke36Ep6/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1598fb28d6a46919e483d60b92f0b8b98df839d1b93f595110bafa7a068a2010","last_reissued_at":"2026-05-18T00:59:58.448770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:58.448770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sensitivity, proximal extension and higher order almost automorphy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Tao Yu, Xiangdong Ye","submitted_at":"2016-05-04T00:17:20Z","abstract_excerpt":"Let $(X,T)$ be a topological dynamical system, and $\\mathcal{F}$ be a family of subsets of $\\mathbb{Z}_+$. $(X,T)$ is strongly $\\mathcal{F}$-sensitive, if there is $\\delta>0$ such that for each non-empty open subset $U$, there are $x,y\\in U$ with $\\{n\\in\\mathbb{Z}_+: d(T^nx,T^ny)>\\delta\\}\\in\\mathcal{F}$. Let $\\mathcal{F}_t$ (resp. $\\mathcal{F}_{ip}$, $\\mathcal{F}_{fip}$) be consisting of thick sets (resp. IP-sets, subsets containing arbitrarily long finite IP-sets).\n  The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either strongly $\\mathcal{F}_{fip"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01119","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":"1605.01119","created_at":"2026-05-18T00:59:58.448843+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.01119v2","created_at":"2026-05-18T00:59:58.448843+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01119","created_at":"2026-05-18T00:59:58.448843+00:00"},{"alias_kind":"pith_short_12","alias_value":"CWMPWKGWURUR","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"CWMPWKGWURURTZED","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"CWMPWKGW","created_at":"2026-05-18T12:30:09.641336+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/CWMPWKGWURURTZED2YFZF4FYXG","json":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG.json","graph_json":"https://pith.science/api/pith-number/CWMPWKGWURURTZED2YFZF4FYXG/graph.json","events_json":"https://pith.science/api/pith-number/CWMPWKGWURURTZED2YFZF4FYXG/events.json","paper":"https://pith.science/paper/CWMPWKGW"},"agent_actions":{"view_html":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG","download_json":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG.json","view_paper":"https://pith.science/paper/CWMPWKGW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.01119&json=true","fetch_graph":"https://pith.science/api/pith-number/CWMPWKGWURURTZED2YFZF4FYXG/graph.json","fetch_events":"https://pith.science/api/pith-number/CWMPWKGWURURTZED2YFZF4FYXG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG/action/storage_attestation","attest_author":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG/action/author_attestation","sign_citation":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG/action/citation_signature","submit_replication":"https://pith.science/pith/CWMPWKGWURURTZED2YFZF4FYXG/action/replication_record"}},"created_at":"2026-05-18T00:59:58.448843+00:00","updated_at":"2026-05-18T00:59:58.448843+00:00"}