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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","authors_text":"Qiang Huo, Xiangtong Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-08-10T15:06:26Z","title":"Entropies of compact subsets and supported measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09702","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:31423bed452f6c31459e3b48ca186209af3a70cdd1141c01dd7872db9231792b","target":"record","created_at":"2026-08-11T02:24:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"548eac4092f6c0b8bb87634c643e583494242f1664c76e6f7de79c8f0b09cfe0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-08-10T15:06:26Z","title_canon_sha256":"be19e82b280d56b3695fcb01e63a2455a2919ecfc0f140abf36c8765f6617838"},"schema_version":"1.0","source":{"id":"2608.09702","kind":"arxiv","version":1}},"canonical_sha256":"8609e462ca8aa45bae5b3d1b1e9772d2fd2ed96afabb3299da9b03a8fdfd9c9c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8609e462ca8aa45bae5b3d1b1e9772d2fd2ed96afabb3299da9b03a8fdfd9c9c","first_computed_at":"2026-08-11T02:24:50.361708Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-11T02:24:50.361708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hheup8DUxPa3478hlVSsPooMDa5cl4gu0s6B10uqil5Vv+eeGSLT9OWXTYsbx9o9PaeL7rq4pOrn5nesJcYHCA==","signature_status":"signed_v1","signed_at":"2026-08-11T02:24:50.363270Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.09702","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31423bed452f6c31459e3b48ca186209af3a70cdd1141c01dd7872db9231792b","sha256:be085789632ae3de7dd87a494f731dee79f394b8220204d2934401daf2587b90"],"state_sha256":"94237973bea48d2ebd360ab2a8bb30a9455591ac493f389fa7823fa92b747344"}