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It is natural to ask if these summands may be further decomposed into equivariant subspaces; that is, if the Oseledets subspaces are reducible. We prove a theorem yielding sufficient conditions for irreducibility of the trivial equivariant subspaces $\\mathbb{R}^2$ and $\\mathbb{C}^2$ for $O"},"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":"1606.02209","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-06-07T16:51:34Z","cross_cats_sorted":[],"title_canon_sha256":"bb81befb9f1c047ec359210dcf55d919b5f3af705e1da18655144e3b2fd5aa18","abstract_canon_sha256":"58c701d9b2896c164f6a5601c8caf756db98215e38d598fef2037f3872c396fb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:01.285302Z","signature_b64":"hRXYYd4D3I5VE+kM0yYTSKyN4qlnSk3tDOZ0dPX+5i4+NbZReFnEZ6zJ2nqjG9HTrsIg5fw+2hDdWFJWmCFdAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01fd864ef0dfcbf8768057a3cf113c9ebae8eb44cd169687684a7f1bdf47ed11","last_reissued_at":"2026-05-18T00:51:01.284899Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:01.284899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On irreducibility of Oseledets subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Anthony Quas, Christopher Bose, Joseph Horan","submitted_at":"2016-06-07T16:51:34Z","abstract_excerpt":"For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of $\\mathbb{R}^n$; that is, above each point in the base space, $\\mathbb{R}^n$ is written as a direct sum of equivariant subspaces, one for each Lyapunov exponent of the cocycle. 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