{"paper":{"title":"Measure-theoretic mean equicontinuity and bounded complexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Tao Yu","submitted_at":"2018-07-13T03:00:02Z","abstract_excerpt":"Let $(X,\\mathcal{B},\\mu,T)$ be a measure preserving system. We say that a function $f\\in L^2(X,\\mu)$ is $\\mu$-mean equicontinuous if for any $\\epsilon>0$ there is $k\\in \\mathbb{N}$ and measurable sets ${A_1,A_2,\\cdots,A_k}$ with $\\mu\\left(\\bigcup\\limits_{i=1}^k A_i\\right)>1-\\epsilon$ such that whenever $x,y\\in A_i$ for some $1\\leq i\\leq k$, one has \\[ \\limsup_{n\\to\\infty}\\frac{1}{n}\\sum_{j=0}^{n-1}|f(T^jx)-f(T^jy)|<\\epsilon. \\] Measure complexity with respect to $f$ is also introduced. It is shown that $f$ is an almost periodic function if and only if $f$ is $\\mu$-mean equicontinuous if and on"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.05868","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"}