{"ID":25121888,"CreatedAt":"2026-09-20T12:27:23.499466772Z","UpdatedAt":"2026-09-20T16:46:37.815414166Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/1102.4635","arxiv_id":"1102.4635","title":"Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton","abstract":"In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us to make some sharp statements concerning the distribution, large-scale geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits. For instance, one of our results is that the union of the unbounded orbits has Hausdorff dimension 1. We give a computer-aided proof of the results concerning the compactification and the renormalization. This proof involves finitely many calculations done with exact integer arithmetic.","short_abstract":"In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us...","url_abs":"https://arxiv.org/abs/1102.4635","url_pdf":"https://arxiv.org/pdf/1102.4635v1","authors":"[\"Richard Evan Schwartz\"]","published":"2011-02-22T22:35:48Z","proceeding":"math.DS","tasks":"[\"math.DS\"]","methods":"[]","has_code":false}