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We show that:\n  \\begin{equation*}\n  \\begin{aligned}\n  & h_{\\mathrm{top}}^{\\mathrm{UC}}(T,K)>0 \\iff h_{\\mathrm{top}}^{\\mathrm{UC}}(T_*,\\mathcal{M}(K))=\\infty, \\qquad\n  &h_{\\mathrm{top}}^{P}(T,K)>0\\iff h_{\\mathrm{top}}^{P}(T_*,\\mathcal{M}(K))>0, \\qquad"},"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":"2608.09702","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-08-10T15:06:26Z","cross_cats_sorted":[],"title_canon_sha256":"be19e82b280d56b3695fcb01e63a2455a2919ecfc0f140abf36c8765f6617838","abstract_canon_sha256":"548eac4092f6c0b8bb87634c643e583494242f1664c76e6f7de79c8f0b09cfe0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T02:24:50.363270Z","signature_b64":"hheup8DUxPa3478hlVSsPooMDa5cl4gu0s6B10uqil5Vv+eeGSLT9OWXTYsbx9o9PaeL7rq4pOrn5nesJcYHCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8609e462ca8aa45bae5b3d1b1e9772d2fd2ed96afabb3299da9b03a8fdfd9c9c","last_reissued_at":"2026-08-11T02:24:50.361708Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T02:24:50.361708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Entropies of compact subsets and supported measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Qiang Huo, Xiangtong Wang","submitted_at":"2026-08-10T15:06:26Z","abstract_excerpt":"Let $(X,T)$ be a topological dynamical system and $(\\mathcal M(X),T_*)$ be its induced system. For a non-empty compact subset $K\\subset X$, we define $\\mathcal M(K)$ as the set of Borel probability measures supported on $K$.\n  In this paper, we systematically study the relationship between various entropies of $(T,K)$ and of $(T_*,\\mathcal M(K))$. We show that:\n  \\begin{equation*}\n  \\begin{aligned}\n  & h_{\\mathrm{top}}^{\\mathrm{UC}}(T,K)>0 \\iff h_{\\mathrm{top}}^{\\mathrm{UC}}(T_*,\\mathcal{M}(K))=\\infty, \\qquad\n  &h_{\\mathrm{top}}^{P}(T,K)>0\\iff h_{\\mathrm{top}}^{P}(T_*,\\mathcal{M}(K))>0, \\qquad"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09702","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.09702/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2608.09702","created_at":"2026-08-11T02:24:50.362382+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.09702v1","created_at":"2026-08-11T02:24:50.362382+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.09702","created_at":"2026-08-11T02:24:50.362382+00:00"},{"alias_kind":"pith_short_12","alias_value":"QYE6IYWKRKSF","created_at":"2026-08-11T02:24:50.362382+00:00"},{"alias_kind":"pith_short_16","alias_value":"QYE6IYWKRKSFXLS3","created_at":"2026-08-11T02:24:50.362382+00:00"},{"alias_kind":"pith_short_8","alias_value":"QYE6IYWK","created_at":"2026-08-11T02:24:50.362382+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/QYE6IYWKRKSFXLS3HUNR5F3S2L","json":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L.json","graph_json":"https://pith.science/api/pith-number/QYE6IYWKRKSFXLS3HUNR5F3S2L/graph.json","events_json":"https://pith.science/api/pith-number/QYE6IYWKRKSFXLS3HUNR5F3S2L/events.json","paper":"https://pith.science/paper/QYE6IYWK"},"agent_actions":{"view_html":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L","download_json":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L.json","view_paper":"https://pith.science/paper/QYE6IYWK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.09702&json=true","fetch_graph":"https://pith.science/api/pith-number/QYE6IYWKRKSFXLS3HUNR5F3S2L/graph.json","fetch_events":"https://pith.science/api/pith-number/QYE6IYWKRKSFXLS3HUNR5F3S2L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L/action/storage_attestation","attest_author":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L/action/author_attestation","sign_citation":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L/action/citation_signature","submit_replication":"https://pith.science/pith/QYE6IYWKRKSFXLS3HUNR5F3S2L/action/replication_record"}},"created_at":"2026-08-11T02:24:50.362382+00:00","updated_at":"2026-08-11T02:24:50.362382+00:00"}