{"paper":{"title":"Portraits of preperiodic points for rational maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.DS"],"primary_cat":"math.NT","authors_text":"Dragos Ghioca, Khoa Nguyen, Thomas J. Tucker","submitted_at":"2014-07-07T03:54:13Z","abstract_excerpt":"Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $\\varphi\\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $\\varphi$ is totally ramified, and let $\\alpha\\in K$. We show that for all but finitely many pairs $(m,n)\\in \\mathbb{Z}_{\\ge 0}\\times \\mathbb{N}$ there exists a place $\\mathfrak{p}$ of $K$ such that the point $\\alpha$ has preperiod $m$ and minimum period $n$ under the action of $\\varphi$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1573","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"}