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We prove that open and densely in $AC_D(M,SL(2,\\mathbb R))$ (in appropriate topologies) the pair $(f,A)$ has simple spectrum with respect to the unique maximal entropy measure $\\mu_f$. On the other hand, there exists a residual subset $\\mathcal{R}\\subset \\text{Aut}_{Leb}(M)\\times L^\\infty(M,SL(2,\\mathbb R))$, with respect to the separate topology, such that any element $(f"},"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":"1209.2293","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-09-11T11:53:09Z","cross_cats_sorted":[],"title_canon_sha256":"81588e1dfb43b33806b83577959b3153799e74ba4234889f70c6b47621b4aef4","abstract_canon_sha256":"0c8761236acf944580e9328f3fde7041bc0507cf57691fe77995eada9b2424cc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:33.662667Z","signature_b64":"bC9hsBbgcRf4Ld2A229rEWGZPthBAvARPg7oRNLPQ8AGKp9GhLB6qnng3+NiSzxfvQnjRMdi70GKEbuobISWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1d997b3fbd0d3868edb717c78a245b68f67f47707205efdacc2ce5aba8bfb06","last_reissued_at":"2026-05-18T03:21:33.662171Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:33.662171Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Mario Bessa, Paulo Varandas","submitted_at":"2012-09-11T11:53:09Z","abstract_excerpt":"Let $AC_D(M,SL(2,\\mathbb R))$ denote the pairs $(f,A)$ so that $f\\in \\mathcal A\\subset \\text{Diff}^{1}(M)$ is a $C^{1}$-Anosov transitive diffeomorphisms and $A$ is an $SL(2,\\mathbb R)$ cocycle dominated with respect to $f$. We prove that open and densely in $AC_D(M,SL(2,\\mathbb R))$ (in appropriate topologies) the pair $(f,A)$ has simple spectrum with respect to the unique maximal entropy measure $\\mu_f$. On the other hand, there exists a residual subset $\\mathcal{R}\\subset \\text{Aut}_{Leb}(M)\\times L^\\infty(M,SL(2,\\mathbb R))$, with respect to the separate topology, such that any element $(f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2293","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1209.2293","created_at":"2026-05-18T03:21:33.662242+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.2293v3","created_at":"2026-05-18T03:21:33.662242+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2293","created_at":"2026-05-18T03:21:33.662242+00:00"},{"alias_kind":"pith_short_12","alias_value":"UHMZPM732DJY","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"UHMZPM732DJYNDW3","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"UHMZPM73","created_at":"2026-05-18T12:27:23.164592+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2","json":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2.json","graph_json":"https://pith.science/api/pith-number/UHMZPM732DJYNDW3OF6HRISFW2/graph.json","events_json":"https://pith.science/api/pith-number/UHMZPM732DJYNDW3OF6HRISFW2/events.json","paper":"https://pith.science/paper/UHMZPM73"},"agent_actions":{"view_html":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2","download_json":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2.json","view_paper":"https://pith.science/paper/UHMZPM73","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.2293&json=true","fetch_graph":"https://pith.science/api/pith-number/UHMZPM732DJYNDW3OF6HRISFW2/graph.json","fetch_events":"https://pith.science/api/pith-number/UHMZPM732DJYNDW3OF6HRISFW2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2/action/storage_attestation","attest_author":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2/action/author_attestation","sign_citation":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2/action/citation_signature","submit_replication":"https://pith.science/pith/UHMZPM732DJYNDW3OF6HRISFW2/action/replication_record"}},"created_at":"2026-05-18T03:21:33.662242+00:00","updated_at":"2026-05-18T03:21:33.662242+00:00"}