{"paper":{"title":"Continuous self-similarity in parametric piecewise isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Franco Vivaldi, John H Lowenstein","submitted_at":"2015-08-24T17:33:05Z","abstract_excerpt":"We exhibit two distinct renormalization scenarios in many-parameter families of piecewise isometries (PWI) of a rhombus. The rotational component, defined over the quadratic field $\\mathbb{K}=\\mathbb{Q}(\\sqrt{5})$, is fixed. The translations are specified by affine functions of the parameters, with coefficients in $\\mathbb{K}$. In each case the parameters range over a convex domain. In one scenario the PWI is self-similar if and only if one parameter belongs to $\\mathbb{K}$, while the other is free. Such a continuous self-similarity is due to the possibility of merging adjacent atoms of an ind"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05885","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"}