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Denote by $Sh(F)$ the space of smooth maps $h:M\\to M$ of the following form: $h(x) = F_{f(x)}(x)$, where $f:M\\to\\mathbb{R}$ runs over all smooth functions on $M$ which can be substituted into the flow $F_t$ instead of time. This space often coincides with the identity component of the group of diffeomorphisms preserving orbits of $F$. In this note it is shown that $Sh(F)$ is not changed under reparametrizations and pushforwards of $F$. As an application it is proved that a vector field $F$ wi"},"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":"0907.0354","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-07-02T12:07:08Z","cross_cats_sorted":[],"title_canon_sha256":"756dd2d1ae932701a47fa8b46a21e8426e53e6e45624b39fcfbe25596eadf244","abstract_canon_sha256":"ece30346f3c575a379eeb735d47ba29b05a2d0461739128cffc97d798240c542"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:48.554054Z","signature_b64":"1ACYTUdmAaj8skENRG2TvzTES9JEhrfuyzh1+FSZOMuJVEFtpPKVVWlhT4n81M14/js+g8hAkrXzZUvmgYmHBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"030ebf0a06acf329f196fa9575277a6bb849397ef359e2de5b9b78bda1c76107","last_reissued_at":"2026-05-18T01:23:48.553359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:48.553359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reparametrizations of vector fields and their shift maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Sergiy Maksymenko","submitted_at":"2009-07-02T12:07:08Z","abstract_excerpt":"Let $M$ be a smooth manifold, $F$ be a smooth vector field on $M$, and $F_t$ be the local flow of $F$. 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