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We show in this paper that, generically, a non-trivial dominated splitting implies partial hyperbolicity for an aperiodic class if it is Lyapunov stable. More precisely, for $C^1$-generic diffeomorphisms, if a Lyapunov stable aperiodic class has a non-trivial dominated splitting $E\\oplus F$, then one of the two bundles is hyperbolic (either $E$ is co"},"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":"1506.07784","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-25T15:20:56Z","cross_cats_sorted":[],"title_canon_sha256":"9b47d0f1c7744f43231f804952905872c719bcca4e6fc278545ba220d1e85b24","abstract_canon_sha256":"f10ed03d88ec4745c7c6985c179e6138974767d93f3b805b5c43f26044ac9055"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:50.816353Z","signature_b64":"9lkPeVFC0aqa7pt8Ztv8s37JrWVerpEye7HpzOvnnhWdIh9L5zkrHZXFGPaZPYUn4HQM+L/bUNrhpDrF8lqlBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05e86bb4277e29c984846e7f0977103e6fbf6dc87c30b605d322d9f24f418565","last_reissued_at":"2026-05-18T01:38:50.815837Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:50.815837Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the dominated splitting of Lyapunov stable aperiodic classes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Xiaodong Wang","submitted_at":"2015-06-25T15:20:56Z","abstract_excerpt":"Recent works related to Palis conjecture of J. Yang, S. Crovisier, M. Sambarino and D. Yang showed that any aperiodic class of a $C^1$-generic diffeomorphism far away from homoclinic bifurcations (or homoclinic tangencies) is partially hyperbolic. We show in this paper that, generically, a non-trivial dominated splitting implies partial hyperbolicity for an aperiodic class if it is Lyapunov stable. 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