{"paper":{"title":"Piecewise contractions and b-adic expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Benito Pires","submitted_at":"2019-06-08T03:36:14Z","abstract_excerpt":"Let $I=[0,1)$, $b\\in \\{2,3,\\ldots\\}$ and $f:I\\to I$ be an injective piecewise $\\frac{1}{b}$-affine map, that is, assume that there exists a partition of $I$ into intervals $I_1,\\ldots,I_n$ such that $\\vert f(x)-f(y)\\vert\\le\\frac1b \\vert x-y\\vert$ for all $x,y\\in I_i$ and $1\\le i\\le n$. In this note, we study the $\\delta$-parameter family of maps $f_{\\delta}=R_{\\delta}\\circ f$, where $R_\\delta:x\\mapsto \\{x+\\delta\\}$. More precisely, we show that the set $\\mathcal{N}$ of parameters $\\delta$ for which $f_{\\delta}$ has only natural codings with maximal complexity is a non-empty set with Hausdorff "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.03382","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"}