{"paper":{"title":"When will a One Parameter Family of Unimodal Maps Produce Finite Limit Cycles Monotonically with the Parameter?","license":"","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"John Taylor","submitted_at":"2007-09-11T11:55:21Z","abstract_excerpt":"In this note we consider a collection $\\cal{C}$ of one parameter families of unimodal maps of $[0,1].$ Each family in the collection has the form $\\{\\mu f\\}$ where $\\mu\\in [0,1].$ Denoting the kneading sequence of $\\mu f$ by $K(\\mu f)$, we will prove that for each member of $\\cal{C}$, the map $\\mu\\mapsto K(\\mu f)$ is monotone. It then follows that for each member of $\\cal{C}$ the map $\\mu\\mapsto h(\\mu f)$ is monotone, where $h(\\u{f})$ is the topological entropy of $\\mu f.$ For interest, $\\mu f(x)=4\\mu x(1-x)$ and $\\mu f(x)=\\mu\\sin(\\pi x)$ are shown to belong to $\\cal{C}.$ This extends the work"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.1595","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"}