{"paper":{"title":"Boundaries of the Arnol'd tongues and the standard family","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Kuntal Banerjee","submitted_at":"2014-02-19T18:19:42Z","abstract_excerpt":"For a family $(F_{t,a} : x \\mapsto x + t + a\\phi(x))$ of increasing homeomorphisms of $\\mathbb R$ with $\\phi$ being Lipschitz continuous of period 1, there is a parameter space consisting of the values $(t,a)$ such that the map $F_{t,a}$ is strictly increasing and it induces an orientation preserving circle homeomorphism. For each $\\theta \\in \\mathbb R$ there is an \\textsf{Arnol'd tongue} $\\mathcal T_\\theta$ of \\textsf{translation number} $\\theta$ in the parameter space. Given a rational $p/q$, it is shown that the boundary $\\partial \\mathcal T_{p/q}$ is a union of two Lipschitz curves which i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4756","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"}