{"paper":{"title":"Dynamics and periodicity in a family of cluster maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Helena Mena-Matos, In\\^es Cruz, M. Esmeralda Sousa-Dias","submitted_at":"2015-11-23T16:05:09Z","abstract_excerpt":"The dynamics of a 1-parameter family of cluster maps $\\varphi_r$ associated to mutation-periodic quivers in dimension 4, is studied in detail. The use of presymplectic reduction leads to a globally periodic symplectic map, and this enables us to reduce the problem to the study of maps belonging to a group of symplectic birational maps of the plane which is isomorphic to $SL(2,\\mathbb{Z})\\ltimes\\mathbb{R}^2$. We conclude that there are three different types of dynamical behaviour for $\\varphi_r$ characterized by the integer parameter values $r=1$, $r=2$ and $r>2$. For each type, the periodic po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07291","kind":"arxiv","version":2},"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"}