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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1906.03382","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-06-08T03:36:14Z","cross_cats_sorted":[],"title_canon_sha256":"e64d1ee32c5703bc989d54521a823eaa9f98da4af47099a661c6ca6449f3738b","abstract_canon_sha256":"4c3a093ccbbf2dd2363ff673161fc6913ed1860a303d64271f73b772511efe8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:48.645990Z","signature_b64":"wYHd+JbY8lxaFa03GToJR9O/kd47ofBHvw4fdRELRArJLzLz34HkdWHOS7ru9PeuZjQiiOi0jEMc5FU7Iky+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ea79112a4080ac70580add292fe83ebf334dc8ce5b653f537cb72e91e4426ad","last_reissued_at":"2026-05-17T23:43:48.645257Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:48.645257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $1\\le i\\le n$. 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