{"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"}