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Given a Lipschitz continuous observable A: {1,2,..,r}^{Z} \\to {R} , we define the map {G}^+: {H}\\to {H} by {G}^+(\\phi)({y}^*) = \\sup_{\\mu \\in {M}_\\sigma} [\\int_{\\{1,2,..,r\\}^{N}} ( A({y}^*, {x}) + \\phi({x})) d\\mu({x}) + h_\\mu(\\sigma) ], where: \\sigma is the left shift map acting on {1,2,..,r}^{N}; {M}_\\sigma denotes the set of \\sigma-invariant Borel probabilities; h_\\mu(\\sigma) indicates the Kolmogorov-Sinai entropy; {H } is the Banach space of Lipschitz real-valued functions on {1,2,..,r}^{N}. We show there "},"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":"1008.1042","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2010-08-05T19:11:14Z","cross_cats_sorted":["cond-mat.stat-mech","math-ph","math.MP","math.PR"],"title_canon_sha256":"b84a58bc97904a3021a31381e1cf84f8d81a651f91ae553ef95bcaa9b3246af8","abstract_canon_sha256":"4e5812ed4c773fa882724b47c2c56936ffc1eddf5a14c9c419a90f5507f04548"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:34.712228Z","signature_b64":"FTCt87QM2G0ePq/zubQNJQ3g7S/wDnafoGf8yHRVmZTU5FKgrV9N7vATpYWf6G1S/C3LLIzD8h3YOLLIXW18Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f835121309e6bb21682f32e976b333eea00b77f0cedf429dca36df39c98ce6c","last_reissued_at":"2026-05-18T04:42:34.711434Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:34.711434Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The effective potential and transshipment in thermodynamic formalism at temperature zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.stat-mech","math-ph","math.MP","math.PR"],"primary_cat":"math.DS","authors_text":"Artur O. Lopes, Eduardo Garibaldi","submitted_at":"2010-08-05T19:11:14Z","abstract_excerpt":"Denote the points in {1,2,..,r}^{Z}= {1,2,..,r}^{N} x {1,2,..,r}^{N} by ({y}^*, {x}). Given a Lipschitz continuous observable A: {1,2,..,r}^{Z} \\to {R} , we define the map {G}^+: {H}\\to {H} by {G}^+(\\phi)({y}^*) = \\sup_{\\mu \\in {M}_\\sigma} [\\int_{\\{1,2,..,r\\}^{N}} ( A({y}^*, {x}) + \\phi({x})) d\\mu({x}) + h_\\mu(\\sigma) ], where: \\sigma is the left shift map acting on {1,2,..,r}^{N}; {M}_\\sigma denotes the set of \\sigma-invariant Borel probabilities; h_\\mu(\\sigma) indicates the Kolmogorov-Sinai entropy; {H } is the Banach space of Lipschitz real-valued functions on {1,2,..,r}^{N}. 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