2004/06/24 by Thierry Monteil, Monteil, Thierry
Computer Science · Engineering · Mathematics · Physics and Astronomy · #32G15 #37C27 #37D50 #37E35 #51E21 #57M12 #Advanced Research in Systems and Signal Processing #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Surface and Thin Film Phenomena #math.DS #msc:32G15 #msc:37C27 #msc:37D50 #msc:37E35 #msc:51E21 #msc:57M12
paper · pdf · doi:10.48550/arxiv.math/0406506
16 pages, 6 figures. v4 : minor changes to take referee's suggestions into account. In particular, an appendix is added with a proof of the following result: "In genus $g\geq 2$, the set of completely periodic translation surfaces has measure zero in every stratum"
openalex publication_date 2004/06/24 · arxiv created 2008/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A translation surface S is said to have the finite blocking property if for every pair (O,A) of points in S there exists a finite number of "blocking" points B1,...,Bn such that every geodesic from O to A meets one of the Bi's. S is said to be purely periodic if the directional flow is periodic in each direction whose directional flow contains a periodic trajectory (this implies that S admits a cylinder decomposition in such directions). We will prove that finite blocking property implies pure periodicity. We will also classify the surfaces that have the finite blocking property in genus 2: such surfaces are exactly the torus branched coverings. Moreover, we prove that in every stratum, such surfaces form a set of null measure. In the Appendix, we prove that completely periodic translation surfaces form a set of null measure in every stratum.