{"paper":{"title":"On proximality with Banach density one","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jian Li, Siming Tu","submitted_at":"2013-12-17T07:21:07Z","abstract_excerpt":"Let $(X,T)$ be a topological dynamical system. A pair of points $(x,y)\\in X^2$ is called Banach proximal if for any $\\epsilon>0$, the set $\\{n\\in\\mathbb{Z}:\\ d(T^nx,T^ny)<\\epsilon\\}$ has Banach density one. We study the structure of the Banach proximal relation. An useful tool is the notion of the support of a topological dynamical system. We show that a dynamical system is strongly proximal if and only if every pair in $X^2$ is Banach proximal. A subset $S$ of $X$ is Banach scrambled if every two distinct points in $S$ form a Banach proximal pair but not asymptotic. We construct a dynamical s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.4668","kind":"arxiv","version":2},"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"}