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For any non-empty compact subset $K$ of $X$, we show that $$\\htop^B(T, K)= \\sup \\{\\underline{h}_\\mu(T): \\mu\\in M(X),\\; \\mu(K)=1\\}, $$ $$\\htop^P(T, K)= \\sup \\{\\bar{h}_\\mu(T): \\mu\\in M(X),\\; \\mu(K)=1\\}. $$ where $\\htop^B(T, K)$ denotes Bowen's topological entropy of $K$, and $\\htop^P(T, K)$ the packing topological entropy of $K$. Furthermore, when $\\h"},"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":"1012.1103","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-12-06T09:38:54Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"66416e8276d88eddd15bc46c69459a5b41482833ad81bf0cc0223b60360faff2","abstract_canon_sha256":"933f9188738aaf6b2a32c7bce045c6218738f9582871af610f2023be420b2c7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:01.815778Z","signature_b64":"Ob5PyB7b1db3umnwZDHwSg+8TawDlNNu7pTXNNQrNTTgaF7M/Tl1GZWEedASNK1CGX6GqxS4zGd4xT2eyKzjDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e89951e5c59c7a1ad22bbdb79699fd81f7dc7c938534569f9c7e744c4d2ecde9","last_reissued_at":"2026-05-18T04:34:01.815175Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:01.815175Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Variational principles for topological entropies of subsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.DS","authors_text":"De-Jun Feng, Wen Huang","submitted_at":"2010-12-06T09:38:54Z","abstract_excerpt":"Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\\underline{h}_\\mu(T)$, $\\bar{h}_\\mu(T)$ for any $\\mu\\in M(X)$, where $M(X)$ denotes the collection of all Borel probability measures on $X$. For any non-empty compact subset $K$ of $X$, we show that $$\\htop^B(T, K)= \\sup \\{\\underline{h}_\\mu(T): \\mu\\in M(X),\\; \\mu(K)=1\\}, $$ $$\\htop^P(T, K)= \\sup \\{\\bar{h}_\\mu(T): \\mu\\in M(X),\\; \\mu(K)=1\\}. $$ where $\\htop^B(T, K)$ denotes Bowen's topological entropy of $K$, and $\\htop^P(T, K)$ the packing topological entropy of $K$. Furthermore, when $\\h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1103","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":"1012.1103","created_at":"2026-05-18T04:34:01.815253+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.1103v1","created_at":"2026-05-18T04:34:01.815253+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1103","created_at":"2026-05-18T04:34:01.815253+00:00"},{"alias_kind":"pith_short_12","alias_value":"5CMVDZOFTR5B","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"5CMVDZOFTR5BVURL","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"5CMVDZOF","created_at":"2026-05-18T12:26:04.259169+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/5CMVDZOFTR5BVURLXW3ZNGP5QH","json":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH.json","graph_json":"https://pith.science/api/pith-number/5CMVDZOFTR5BVURLXW3ZNGP5QH/graph.json","events_json":"https://pith.science/api/pith-number/5CMVDZOFTR5BVURLXW3ZNGP5QH/events.json","paper":"https://pith.science/paper/5CMVDZOF"},"agent_actions":{"view_html":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH","download_json":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH.json","view_paper":"https://pith.science/paper/5CMVDZOF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.1103&json=true","fetch_graph":"https://pith.science/api/pith-number/5CMVDZOFTR5BVURLXW3ZNGP5QH/graph.json","fetch_events":"https://pith.science/api/pith-number/5CMVDZOFTR5BVURLXW3ZNGP5QH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH/action/storage_attestation","attest_author":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH/action/author_attestation","sign_citation":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH/action/citation_signature","submit_replication":"https://pith.science/pith/5CMVDZOFTR5BVURLXW3ZNGP5QH/action/replication_record"}},"created_at":"2026-05-18T04:34:01.815253+00:00","updated_at":"2026-05-18T04:34:01.815253+00:00"}