{"paper":{"title":"Hofer's $L^{\\infty}$-geometry: energy and stability of Hamiltonian flows, part I","license":"","headline":"","cross_cats":["math.DG"],"primary_cat":"math.DS","authors_text":"Dusa McDuff, Fran\\c{c}ois Lalonde","submitted_at":"1995-03-09T00:00:00Z","abstract_excerpt":"Consider the group $\\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\\om)$ with the Hofer $L^{\\infty}$-norm. A path in $\\Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\\Ll$. In this paper, we give a necessary condition for a path $\\ga$ to be a geodesic. We also develop a necessary condition for a geodesic to be stable, that is, a local minimum for $\\Ll$. This condition is related to the existence of periodic orbits for the linearization of the path, and so extends Ustilo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9503227","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"}