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Consider a set of points with non-dense forward orbit: $$E(f, y) := \\{ z\\in M: y\\notin \\overline{\\{f^k(z), k \\in \\mathbb{N}\\}}\\}$$ for some $y \\in M$ and $$E_{x}(f, y) := E(f, y) \\cap W^u(x)$$ for any $x\\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimens"},"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":"1504.01835","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-04-08T06:03:04Z","cross_cats_sorted":[],"title_canon_sha256":"22bda107070750d4bdc504ac6eb714212768b1a5ee74959e2bc0a0db7fb0d330","abstract_canon_sha256":"9e84594b9cae0d11b4734708f759d89ebf17c9c75a2a2cfb5c1f0a52db85885e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:00.079138Z","signature_b64":"0qczsbO7OT+0QoS3FkgVokqUq5S9GBjQO4/vpFqCmIK7PUjtofU7wvCA8EMT9CNtu6FcIohZ0Ad/P4BM6s1QCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"259a179fad2ebe9eeea5564201111540e89375eef49ee754ade561b489812157","last_reissued_at":"2026-05-18T02:19:00.078594Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:00.078594Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Weisheng Wu","submitted_at":"2015-04-08T06:03:04Z","abstract_excerpt":"Let $f: M \\to M$ be a $C^{1+\\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of \\cite{Wu} as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) := \\{ z\\in M: y\\notin \\overline{\\{f^k(z), k \\in \\mathbb{N}\\}}\\}$$ for some $y \\in M$ and $$E_{x}(f, y) := E(f, y) \\cap W^u(x)$$ for any $x\\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01835","kind":"arxiv","version":2},"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":"1504.01835","created_at":"2026-05-18T02:19:00.078681+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.01835v2","created_at":"2026-05-18T02:19:00.078681+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.01835","created_at":"2026-05-18T02:19:00.078681+00:00"},{"alias_kind":"pith_short_12","alias_value":"EWNBPH5NF27J","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EWNBPH5NF27J53VF","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EWNBPH5N","created_at":"2026-05-18T12:29:19.899920+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/EWNBPH5NF27J53VFKZBACEIVID","json":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID.json","graph_json":"https://pith.science/api/pith-number/EWNBPH5NF27J53VFKZBACEIVID/graph.json","events_json":"https://pith.science/api/pith-number/EWNBPH5NF27J53VFKZBACEIVID/events.json","paper":"https://pith.science/paper/EWNBPH5N"},"agent_actions":{"view_html":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID","download_json":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID.json","view_paper":"https://pith.science/paper/EWNBPH5N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.01835&json=true","fetch_graph":"https://pith.science/api/pith-number/EWNBPH5NF27J53VFKZBACEIVID/graph.json","fetch_events":"https://pith.science/api/pith-number/EWNBPH5NF27J53VFKZBACEIVID/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID/action/storage_attestation","attest_author":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID/action/author_attestation","sign_citation":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID/action/citation_signature","submit_replication":"https://pith.science/pith/EWNBPH5NF27J53VFKZBACEIVID/action/replication_record"}},"created_at":"2026-05-18T02:19:00.078681+00:00","updated_at":"2026-05-18T02:19:00.078681+00:00"}