{"paper":{"title":"Questions about the dynamics on a natural family of affine cubic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.AG","authors_text":"Julio Rebelo, Roland Roeder","submitted_at":"2023-07-20T15:40:35Z","abstract_excerpt":"We present several questions about the dynamics of the group of holomorphic automorphisms of the affine cubic surfaces $$S_{A,B,C,D} = \\{(x,y,z) \\in \\mathbb{C}^3 \\, : \\, x^2 + y^2 + z^2 +xyz = Ax + By+Cz+D\\},$$ where $A,B,C,$ and $D$ are complex parameters. This group action describes the monodromy of the famous Painlev\\'e 6 Equation as well as the natural dynamics of the mapping class group on the ${\\rm SL}(2,\\mathbb{C})$ character varieties associated to the once punctured torus and the four times punctured sphere.\n  The questions presented here arose while preparing our work ``Dynamics of g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.10962","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2307.10962/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}