{"paper":{"title":"The Octagonal PET I: Renormalization and Hyperbolic Symmetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Richard Evan Schwartz","submitted_at":"2012-09-11T18:37:31Z","abstract_excerpt":"We introduce a family of polytope exchange transformations (PETs) acting on parallelotopes in $\\R^{2n}$ for $n=1,2,3...$. These PETs are constructed using a pair of lattices in $\\R^{2n}$. The moduli space of these PETs is $GL_n(\\R)$. We study the case n=1 in detail. In this case, we show that the 2-dimensional family is completely renormalizable and that the $(2,4,\\infty)$ hyperbolic reflection triangle group acts (by linear fractional transformations) as the renormalization group on the moduli space. These results have a number of geometric corollaries for the system. Most of the paper is tra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2390","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"}