Geodesics on flat surfaces

Abstract

Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. In an important particular case, when the flat metric has trivial holonomy, the corresponding flat surfaces are naturally organized into families which appear to be isomorphic to moduli spaces of holomorphic oneforms.

One can obtain much information about the geometry and dynamics of an individual flat surface by studying both its orbit under the Teichmüller geodesic flow and under the linear group action on the corresponding moduli space. We apply this general principle to the study of generic geodesics and to counting of closed geodesics on a flat surface.