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We construct appropriately normalized canonical heights h_0,h_1,h_2,... associated to D_0,D_1,D_2,... and satisfying Jordan transformation formulas h_k(f(x)) = b h_k(x) + h_{k-1}(x). As an application, we prove that for every x in X, the arithmetic degree a_f(x) exists, is an algebraic integer, and takes on only finitely many values as x varies over X. Further, if X"},"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":"1301.4964","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-01-21T19:14:37Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"13493fac826bda118ae6efa376195131615284be84f87b3d1dd3a82f26fc88ae","abstract_canon_sha256":"830f1a455ed6dd2695fa8b7dcade60e8eb348964dce88b85d0441b38d025b961"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:36.021137Z","signature_b64":"D/kNTMJ/VCppoRSKx+TK69jD7r0D8zxBkeNAZ+/bqPI52RtlEiwt8tORnXGVQd202Y6e2TbqiZjJOtAazLVnBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff30efbccb56f1749832ac0c4269e66758abe7a4419d3691123f751647da2cfa","last_reissued_at":"2026-05-18T00:36:36.020569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:36.020569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Joseph H. Silverman, Shu Kawaguchi","submitted_at":"2013-01-21T19:14:37Z","abstract_excerpt":"Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the action of f^* on Pic(X) tensored with C. We construct appropriately normalized canonical heights h_0,h_1,h_2,... associated to D_0,D_1,D_2,... and satisfying Jordan transformation formulas h_k(f(x)) = b h_k(x) + h_{k-1}(x). As an application, we prove that for every x in X, the arithmetic degree a_f(x) exists, is an algebraic integer, and takes on only finitely many values as x varies over X. 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