{"paper":{"title":"The $C^1$ density of nonuniform hyperbolicity in $C^{ r}$ conservative diffeomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Chao Liang, Yun Yang","submitted_at":"2015-08-27T03:32:13Z","abstract_excerpt":"Let $\\Diff^{ r}_m(M)$ be the set of $C^{ r}$ volume-preserving diffeomorphisms on a compact Riemannian manifold $M$ ($\\dim M\\geq 2$). In this paper, we prove that the diffeomorphisms without zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\\Diff^{ r}_m(M), r\\geq 1$. We also prove a weaker result for symplectic diffeomorphisms $\\mathcal{S}ym^{r}_{\\omega}(M), r\\geq1 $ saying that the symplectic diffeomorphisms with non-zero Lyapunov exponents on a set of positive volume are $C^1$ dense in $\\mathcal{S}ym^{r}_{\\omega}(M), r\\geq1 $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.06714","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"}