{"paper":{"title":"The $\\beta$-transformation with a hole at 0","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Charlene Kalle, Derong Kong, Niels Langeveld, Wenxia Li","submitted_at":"2018-03-20T09:52:19Z","abstract_excerpt":"For $\\beta\\in(1,2]$ the $\\beta$-transformation $T_\\beta: [0,1) \\to [0,1)$ is defined by $T_\\beta ( x) = \\beta x \\pmod 1$. For $t\\in[0, 1)$ let $K_\\beta(t)$ be the survivor set of $T_\\beta$ with hole $(0,t)$ given by \\[K_\\beta(t):=\\{x\\in[0, 1): T_\\beta^n(x)\\not \\in (0, t) \\textrm{ for all }n\\ge 0\\}.\\] In this paper we characterise the bifurcation set $E_\\beta$ of all parameters $t\\in[0,1)$ for which the set valued function $t\\mapsto K_\\beta(t)$ is not locally constant. We show that $E_\\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\\beta\\in(1,2)$. We prove that for Lebesgue a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07338","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"}