{"paper":{"title":"On the sharpness of the $C^1$-norm threshold for perturbations in the normally hyperbolic invariant manifold theorem---a toy model perspective","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Junhao Li, Lin Wang, Yujie Huang","submitted_at":"2026-08-05T13:54:33Z","abstract_excerpt":"The classical normally hyperbolic invariant manifold theorem asserts that a \\(C^1\\) normally hyperbolic invariant manifold persists under \\(C^1\\) small perturbations. For a family of standard-like dissipative twist maps, we show that the threshold \\((1-\\sqrt{\\lambda})^2\\) for the \\(C^1\\)-norm of the perturbation is sharp: there exists a $C^\\infty$ perturbation \\(\\phi\\) with \\(\\|\\phi\\|_{C^1} = (1-\\sqrt{\\lambda})^2\\) such that the map preserves a unique invariant graph, but this graph possesses non-differentiable points. On the other hand, whenever \\(\\|\\phi\\|_{C^1} < (1-\\sqrt{\\lambda})^2\\), the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.04862","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.04862/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}