{"paper":{"title":"An Extended Fatou-Shishikura inequality and wandering branch continua for polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Alexander Blokh, Dierk Schleicher, Doug Childers, Genadi Levin, Lex Oversteegen","submitted_at":"2010-01-06T20:08:33Z","abstract_excerpt":"Let $P$ be a polynomial of degree $d$ with Julia set $J_P$. Let $\\widetilde N$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikura inequality $\\widetilde N\\le d-1$. The goal of the paper is to improve this bound. The new count includes \\emph{wandering collections of non-precritical branch continua}, i.e., collections of continua or points $Q_i\\subset J_P$ \\emph{all} of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of $\\mathrm{eval}(Q_i)\\ge 3$ external rays. Also, we relate individual cycles, which are either non-repelling or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.0953","kind":"arxiv","version":3},"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"}