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We prove that there is a sequence of horseshoes for $f$ and dominated splittings for $\\mathcal{A}$ on the horseshoes, such that not only the measure theoretic entropy of $f$ but also the Lyapunov exponents of $\\mathcal{A}$ with respect to $\\mu$ can be approximated by the topological entropy of $f$ and the Lyapunov exponents of"},"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":"1902.05768","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2019-02-15T11:04:13Z","cross_cats_sorted":[],"title_canon_sha256":"32efd4b79faeafbd0bef867c9c56b045dae3321b986433727ca61c743a3c5792","abstract_canon_sha256":"4abfd5afc556e061d42f1cb0aa23cda07b6bf9638a950d9353c9193c1871f1bc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:56.218037Z","signature_b64":"PzLsnGwTXA4fcVRQ4ydfB6ciX3/iDI5vI3fDa1fhvv2Bh3WA+4qKdGMpZPwiqqNEdocyQzPlTzc6nyst6AjsBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e14b74f277df412fa1123ccea951df9bf0c7bd574552554eec2d874c052705c5","last_reissued_at":"2026-05-17T23:53:56.217363Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:56.217363Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Horseshoes and Lyapunov exponents for Banach cocycles over nonuniformly hyperbolic systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Rui Zou, Yongluo Cao","submitted_at":"2019-02-15T11:04:13Z","abstract_excerpt":"Let $f$ be a $C^r$$(r>1)$ diffeomorphism of a compact Riemannian manifold $M$, preserving an ergodic hyperbolic measure $\\mu$ with positive entropy, and let $\\mathcal{A}$ be a H\\\"older continuous cocycle of injective bounded linear operators acting on a Banach space $X$. 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