{"paper":{"title":"Periods of rational maps modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Benjamin Hutz, Dragos Ghioca, P\\\"ar Kurlberg, Robert L. Benedetto, Thomas J. Tucker, Thomas Scanlon","submitted_at":"2011-07-14T13:35:46Z","abstract_excerpt":"Let $K$ be a number field, let $\\phi \\in K(t)$ be a rational map of degree at least 2, and let $\\alpha, \\beta \\in K$. We show that if $\\alpha$ is not in the forward orbit of $\\beta$, then there is a positive proportion of primes ${\\mathfrak p}$ of $K$ such that $\\alpha \\mod {\\mathfrak p}$ is not in the forward orbit of $\\beta \\mod {\\mathfrak p}$.\n  Moreover, we show that a similar result holds for several maps and several points.\n  We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $\\alpha$ by a hypersurface, su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2816","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"}