{"paper":{"title":"Irregular sets, the $\\beta$-transformation and the almost specification property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniel J. Thompson","submitted_at":"2009-05-06T13:05:31Z","abstract_excerpt":"Let $(X,d)$ be a compact metric space, $f:X \\mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $\\phi$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $\\phi$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $\\beta$-shift satisfies almost specification and we show that the irregular set for any $\\beta$-shift or $\\beta$-transformation is either empty or has full topologic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.0739","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"}