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We also assume $\\log g$ is well defined for this extension.\n  It is known in Complex Dynamics that under the above hypothesis, for the given potential $\\beta \\,\\log g$, where $\\beta$ is a real constant, there exists a real analytic eigenfunction $\\phi_\\beta$ defined on $[0,1]$ (with a complex"},"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":"1205.5758","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-05-25T17:23:07Z","cross_cats_sorted":["math.CV","math.OC"],"title_canon_sha256":"49569842862ae09d618d6f1e869fd8f47bca2ab70ce8f8ac0fd6eaab86b973bc","abstract_canon_sha256":"9d24234716be4c46bc486d242b5343f96853fe51b1feac2f3859c59fbbe8ac5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:50.424498Z","signature_b64":"t0BnikmWJur6BcssMnjEo5tL4G2iL6QpT5YkW3I0Cei8yXU6rQLPin7Tox02wnijCLPzF/PxrEGddvdX45q1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b99a23d5e6b487d477d6eaa0aba303c7dc940adde0f05582cd27387025bc7ab1","last_reissued_at":"2026-05-18T03:54:50.423709Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:50.423709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.OC"],"primary_cat":"math.DS","authors_text":"Artur O. 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