{"paper":{"title":"Topological properties of a class of cubic Rauzy fractals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Beno\\^it Loridant","submitted_at":"2014-11-27T10:53:04Z","abstract_excerpt":"We consider the substitution $\\sigma_{a,b}$ defined by $$\\begin{array}{rlcl} \\sigma_{a,b}: & 1 & \\mapsto & \\underbrace{1\\ldots 1}_{a}2 \\\\ & 2 & \\mapsto & \\underbrace{1\\ldots 1}_{b}3 \\\\ & 3 & \\mapsto & 1 \\end{array} $$ with $a\\geq b\\geq 1$. The shift dynamical system induced by $\\sigma_{a,b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile $\\mathcal{T}_{a,b}$ with fractal boundary.\n  We prove that $\\mathcal{T}_{a,b}$ is homeomorphic to the closed disk iff $2b-a\\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.7544","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"}