{"paper":{"title":"An elementary proof of Franks' lemma for geodesic flows","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Daniel Visscher","submitted_at":"2013-07-24T20:25:43Z","abstract_excerpt":"Given a Riemannian manifold $(M,g)$ and a geodesic $\\gamma$, the perpendicular part of the derivative of the geodesic flow $\\phi_g^t: SM \\rightarrow SM$ along $\\gamma$ is a linear symplectic map. We give an elementary proof of the following Franks' lemma, originally found in [G. Contreras and G. Paternain, 2002] and [G. Contreras, 2010]: this map can be perturbed freely within a neighborhood in $Sp(n)$ by a $C^2$-small perturbation of the metric $g$ that keeps $\\gamma$ a geodesic for the new metric. Moreover, the size of these perturbations is uniform over fixed length geodesics on the manifol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.6573","kind":"arxiv","version":2},"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"}