{"paper":{"title":"On density of positive Lyapunov exponents for $C^1$ symplectic diffeomorphisms","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Chao Liang","submitted_at":"2015-06-17T01:45:36Z","abstract_excerpt":"Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $\\omega$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$ approximated by one with a positive Lyapunov exponent for a positive-measured subset of $M$. That is, the set \\[ \\left\\{ f\\in \\mathcal{S}ym^1_{\\omega}(M)\\,| \\begin{array}{ll} &\\mbox{The largest Lyapunov exponent }\\lambda_1(f,\\,x)>0\\\\ &\\mbox{ for a positive measure set } \\end{array} \\right\\} \\] is dense in $\\mathcal{S}ym^1_{\\omega}(M)$. \\end{abstract} \\end{center"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.05181","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"}