{"paper":{"title":"A lower bound on the orbit growth of a regular self-map of affine space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Vesselin Dimitrov","submitted_at":"2013-11-17T09:16:18Z","abstract_excerpt":"We show that if $f : \\mathbb{A}_{\\bar{\\mathbb{Q}}}^r \\to \\mathbb{A}_{\\bar{\\mathbb{Q}}}^r$ is a regular self-map and $P \\in \\mathbb{A}^r(\\bar{\\mathbb{Q}})$ has $\\limsup_{n \\in \\mathbb{N}} \\frac{\\log{h_{\\mathrm{aff}}(f^nP)}}{\\log{n}} < 1/r$, where $h_{\\textrm{aff}}$ is the affine Weil height, then $\\mathbb{N}$ partitions into a finite set and finitely many full arithmetic progressions, on each of which the coordinates of $f^nP$ are polynomials in $n$.\n  In particular, if $(f^nP)_{n \\in \\mathbb{N}}$ is a Zariski-dense orbit, then either $n = 1$ and $f$ is of the shape $t \\mapsto \\zeta t + c$, $\\z"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4133","kind":"arxiv","version":1},"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"}