{"paper":{"title":"Expansive multiparameter actions and mean dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Masaki Tsukamoto, Tom Meyerovitch","submitted_at":"2017-10-26T11:29:58Z","abstract_excerpt":"Ma\\~n\\'e (1979) proved that if a compact metric space admits an expansive homeomorphism then it is finite dimensional. We generalize this theorem to multiparameter actions. The generalization involves mean dimension theory, which counts \"averaged dimension\" of a dynamical system. We prove that if $T:\\mathbb{Z}^k\\times X\\to X$ is expansive and if $R:\\mathbb{Z}^{k-1}\\times X\\to X$ commutes with $T$ then $R$ has finite mean dimension. When $k=1$, this statement reduces to Ma\\~{n}\\'{e}'s theorem. We also study several related issues, especially the connection with entropy theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09647","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"}