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If $u_\\lambda:M\\to\\mathbb R$ is the viscosity solution of the discounted equation $$\n  \\lambda u_\\lambda(x)+H(x,d_x u_\\lambda)=c(H), $$ where $c(H)$ is the critical value, we prove that $u_\\lambda$ converges uniformly, as $\\lambda\\to 0$, to a specific solution $u_0:M\\to\\mathbb R$ of the critical equation $$ H(x,d_x u)=c(H). $$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1408.6712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-08-28T13:38:34Z","cross_cats_sorted":[],"title_canon_sha256":"e434164223386768b71d801475894c91f4fbca0247b15d8068b7c6150453d1ab","abstract_canon_sha256":"e98efab952d1d1e79e14aac1d8553431e3bba037f4108a3c086b168397b76663"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:05.396561Z","signature_b64":"67+oXojl2MAsbUiKMp6l6jGUgYAat47d+ZUxgO02XAHdHAqUD5TeBS+PhBrWJBYrYLp0sDedywQTKyW552GnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea0d0787d2f57238def54f6987b48ed29f6a843514f31e663f05bc9422569356","last_reissued_at":"2026-05-18T01:21:05.395909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:05.395909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Convergence of the solutions of the discounted equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Albert Fathi, Andrea Davini, Maxime Zavidovique, Renato Iturriaga","submitted_at":"2014-08-28T13:38:34Z","abstract_excerpt":"We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. 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