2016/05/31 by C. Christopher Johnson, Charles C. Johnson
Computer Science · Materials Science · Mathematics · #Combinatorics #Computer science #Geodesic #Geometry #Interval (graph theory) #Mathematical Dynamics and Fractals #Mathematics #Moduli space #Quasicrystal Structures and Properties #Rotation (mathematics) #Sequence (biology) #Skew #Space (punctuation) #Square (algebra) #Surface (topology) #Translation (biology) #math.DS #math.MG #msc:37E35 #semigroups and automata theory
paper · pdf · doi:10.1007/s10711-017-0227-z
28 pages, 12 figures. Minor revisions and corrections. To appear in Geometriae Dedicata
arxiv created 2017/02/02 · arxiv updated 2017/02/06 · openalex publication_date 2017/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals on the surface. These interval exchanges can be thought of as skew products over a rotation, and we convert cutting sequences to symbolic trajectories of these interval exchanges to show that special types of combinatorial lifts of Sturmian sequences completely describe all cutting sequences on a square-tiled surface. Our results extend the list of families of surfaces where cutting sequences are understood to a dense subset of the moduli space of all translation surfaces.