{"paper":{"title":"Examples of nearly integrable systems on $\\mathbb{A}^3$ with asymptotically dense projected orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SG"],"primary_cat":"math.DS","authors_text":"Jean-Pierre Marco, Lara Sabbagh","submitted_at":"2014-01-15T14:02:34Z","abstract_excerpt":"Given an integer $\\kappa\\geq2$, we introduce a class of nearly integrable systems on $\\mathbb{A}^3$, of the form $$ H_n(\\theta,r)=\\frac12 \\Vert r\\Vert ^2+\\tfrac{1}{n} U(\\theta_2,\\theta_3)+f_n(\\theta,r) $$ where $U\\in C^\\kappa(\\mathbb{T}^2)$ is a generic potential function and $f_n$ a $C^{\\kappa-1}$ additional perturbation such that $\\Vert f_n\\Vert_{C^{\\kappa-1}(\\mathbb{A}^3)}\\leq \\tfrac{1}{n}$, so that $H_n$ is a perturbation of the completely integrable system $h(r)=\\frac12\\Vert r\\Vert ^2$.\n  Let $\\Pi:\\mathbb{A}^3\\to\\mathbb{R}^3$ be the canonical projection. We prove that for each $\\delta>0$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3593","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"}