Cutting sequences on square-tiled surfaces

Abstract

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.Comment: 28 pages, 12 figures. Minor revisions and corrections. To appear in Geometriae Dedicat

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

“…For example, in Section 4 we sketch a similar result for cutting sequences. [Joh17]. Beyond the Veech case, there is a long string of papers of Lopez-Narbel developing a language-theoretic formulation of generalized Sturmian sequences for interval exchange transformations, beginning with [LN95].…”