{"paper":{"title":"On Poincar\\'e extensions of rational maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Carlos Cabrera, Guillermo Sienra, Peter Makienko","submitted_at":"2013-05-30T17:10:59Z","abstract_excerpt":"There is a classical extension, of M\\\"obius automorphisms of the Riemann sphere into isometries of the hyperbolic space $\\mathbb{H}^3$, which is called the Poincar\\'e extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of $\\mathbb{H}^3$ exploiting the fact that any holomorphic covering between Riemann surfaces is M\\\"obius for a suitable choice of coordinates.\n  We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a produc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.7164","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"}