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First, we show with some conditions that a $K$-morphism $\\phi: X_K \\to X_K$ of degree at least two is isotrivial if and only if $\\phi$ has potential good reduction at all places $v$ of $K$. Second, let $(X,\\phi), (Y,\\psi)$ be dynamical systems where $X,Y$ are defined over $k$ and $g:X_{K} \\to Y_{K}$ a dominant $K$-morphism, such that $g \\circ \\phi = \\psi \\circ g$. We show under certain conditions that if $\\phi$ is defined over $k$, then"},"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":"1303.6646","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-03-26T20:07:03Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"0677b15dd68225fbb733573987d61132d703d895a7021f5bf21f8918eecb1543","abstract_canon_sha256":"fc38c93e5c5c628f6376b93f0ffceb3460a8e3f17c36cbeade05bc82c50bd456"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:01.931682Z","signature_b64":"WkhcQ9nmftllh32wRg+tljwwJDDS3nm4+ydvZ3bwFMXUN2LNUowo/oZ0XQxUM0Pd0atrv1yHz3HQ/r38nY40AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8fe6ef35c5eabd670d70595b78694d6c9a0954cca1870ced79003f0ea7ebd25b","last_reissued_at":"2026-05-18T03:07:01.931186Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:01.931186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Isotriviality and the space of morphisms on projective varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.DS","authors_text":"Alon Levy, Anupam Bhatnagar","submitted_at":"2013-03-26T20:07:03Z","abstract_excerpt":"Let $K=k(C)$ be the function field of a smooth projective curve $C$ over an infinite field $k$, let $X$ be a projective variety over $k$. We prove two results. First, we show with some conditions that a $K$-morphism $\\phi: X_K \\to X_K$ of degree at least two is isotrivial if and only if $\\phi$ has potential good reduction at all places $v$ of $K$. Second, let $(X,\\phi), (Y,\\psi)$ be dynamical systems where $X,Y$ are defined over $k$ and $g:X_{K} \\to Y_{K}$ a dominant $K$-morphism, such that $g \\circ \\phi = \\psi \\circ g$. 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