{"paper":{"title":"Roots, Schottky semigroups, and a proof of Bandt's Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.GT"],"primary_cat":"math.DS","authors_text":"Alden Walker, Danny Calegari, Sarah Koch","submitted_at":"2014-10-30T20:14:50Z","abstract_excerpt":"In 1985, Barnsley and Harrington defined a ``Mandelbrot Set'' $\\mathcal{M}$ for pairs of similarities --- this is the set of complex numbers $z$ with $0<|z|<1$ for which the limit set of the semigroup generated by the similarities $x \\mapsto zx$ and $x \\mapsto z(x-1)+1$ is connected. Equivalently, $\\mathcal{M}$ is the closure of the set of roots of polynomials with coefficients in $\\lbrace -1,0,1 \\rbrace$. Barnsley and Harrington already noted the (numerically apparent) existence of infinitely many small ``holes'' in $\\mathcal{M}$, and conjectured that these holes were genuine. These holes are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8542","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"}