{"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"}