{"paper":{"title":"Entropy, topological transitivity, and dimensional properties of unique $q$-expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Derong Kong, Rafael Alcaraz Barrera, Simon Baker","submitted_at":"2016-09-07T19:20:56Z","abstract_excerpt":"Let $M$ be a positive integer and $q \\in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\\{0,\\ldots, M\\}$. In particular, we study the set $\\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\\mathbf{U}_q$ of corresponding sequences.\n  It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\\in(1, M+1]$ the topological entropy of $\\mathcal{U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\\mathcal E$ of $q$'s where the f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.02122","kind":"arxiv","version":3},"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"}