{"paper":{"title":"Boundaries of Disk-like Self-affine Tiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.GT"],"primary_cat":"math.MG","authors_text":"Jun Jason Luo, King-Shun Leung","submitted_at":"2012-06-02T13:39:13Z","abstract_excerpt":"Let $T:= T(A, {\\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\\omega$, we find the generalized Hausdorff dimension $\\dim_H^{\\omega} (\\partial T)=2\\log \\rho(M)/\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0382","kind":"arxiv","version":2},"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"}